﻿<?xml version="1.0" encoding="utf-8"?>
<InstanceReport xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  <Version>2.2.0.7</Version>
  <hasSegments>false</hasSegments>
  <ReportName>Debt of the Parent Company</ReportName>
  <ReportLongName>0205 - Disclosure - Debt of the Parent Company</ReportLongName>
  <DisplayLabelColumn>true</DisplayLabelColumn>
  <ShowElementNames>false</ShowElementNames>
  <RoundingOption />
  <HasEmbeddedReports>false</HasEmbeddedReports>
  <Columns>
    <Column>
      <LabelColumn>false</LabelColumn>
      <Id>1</Id>
      <Labels>
        <Label Id="1" Label="6 Months Ended" />
        <Label Id="2" Label="Jun. 30, 2010" />
      </Labels>
      <CurrencyCode />
      <FootnoteIndexer />
      <hasSegments>false</hasSegments>
      <hasScenarios>false</hasScenarios>
      <Segments />
      <Scenarios />
      <Units />
      <CurrencySymbol />
    </Column>
  </Columns>
  <Rows>
    <Row>
      <Id>2</Id>
      <Label>Debt of the Parent Company [Abstract]</Label>
      <Level>0</Level>
      <ElementName>amb_DebtOfParentCompanyAbstract</ElementName>
      <ElementPrefix>amb</ElementPrefix>
      <IsBaseElement>false</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>Debt of the Parent Company Abstract.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>true</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole />
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText />
          <NonNumericTextHeader />
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>xbrli:stringItemType</ElementDataType>
      <SimpleDataType>string</SimpleDataType>
      <ElementDefenition>Debt of the Parent Company Abstract.</ElementDefenition>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
    <Row>
      <Id>3</Id>
      <Label>Debt of the Parent Company</Label>
      <Level>1</Level>
      <ElementName>us-gaap_ScheduleOfGuaranteeObligationsTextBlock</ElementName>
      <ElementPrefix>us-gaap</ElementPrefix>
      <IsBaseElement>true</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>false</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole>verboselabel</PreferredLabelRole>
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText>&lt;!--DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd" --&gt;
   &lt;!-- Begin Block Tagged Note 5 - us-gaap:ScheduleOfGuaranteeObligationsTextBlock--&gt;
   &lt;div style="margin-left: 0%"&gt;
   &lt;div style="margin-top: 12pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;table width="100%" border="0" cellpadding="0" cellspacing="0" style="font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: transparent; text-align: left"&gt;
   &lt;tr&gt;
       &lt;td width="3%"&gt;&lt;/td&gt;
       &lt;td width="97%"&gt;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr valign="top"&gt;
       &lt;td&gt;
       &lt;b&gt;&lt;font style="font-family: 'Times New Roman', Times"&gt;5.&amp;#160;&amp;#160;&lt;/font&gt;&lt;/b&gt;
   &lt;/td&gt;
       &lt;td&gt;
       &lt;b&gt;&lt;font style="font-family: 'Times New Roman', Times"&gt;Debt of
       the Parent Company&lt;/font&gt;&lt;/b&gt;
   &lt;/td&gt;
   &lt;/tr&gt;
   &lt;/table&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company itself does not hold any indebtedness. All
       debt is held directly or indirectly by the Operating
       Partnership. The debt that is guaranteed by the Parent Company
       is discussed below. Note&amp;#160;6 below entitled &amp;#8220;Debt of the
       Operating Partnership&amp;#8221; should be read in conjunction with
       this Note&amp;#160;5 for a discussion of the debt of the Operating
       Partnership consolidated into the Parent Company&amp;#8217;s
       financial statements. In this Note&amp;#160;5, the &amp;#8220;Parent
       Company&amp;#8221; refers only to AMB Property Corporation and not to
       any of its subsidiaries.
   &lt;/div&gt;
   &lt;div style="margin-top: 12pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 2%; margin-right: 0%; font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: #ffffff"&gt;
       &lt;b&gt;&lt;i&gt;&lt;font style="font-family: 'Times New Roman', Times"&gt;Unsecured
       Senior Debt Guarantees&lt;/font&gt;&lt;/i&gt;&lt;/b&gt;
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company guarantees the Operating Partnership&amp;#8217;s
       obligations with respect to its unsecured senior debt
       securities. As of June&amp;#160;30, 2010, the Operating Partnership
       had outstanding an aggregate of $1.2&amp;#160;billion in unsecured
       senior debt securities, which bore a weighted average interest
       rate of 6.4% and had an average term of 5.6&amp;#160;years. The
       indenture for the senior debt securities contains limitations on
       mergers or consolidations of the Parent Company.
   &lt;/div&gt;
   &lt;div style="margin-top: 12pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 2%; margin-right: 0%; font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: #ffffff"&gt;
       &lt;b&gt;&lt;i&gt;&lt;font style="font-family: 'Times New Roman', Times"&gt;Other
       Debt Guarantees&lt;/font&gt;&lt;/i&gt;&lt;/b&gt;
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company guarantees certain of the Operating
       Partnership&amp;#8217;s other debt obligations related to its
       $425.0&amp;#160;million multi-currency senior unsecured term loan
       facility, which includes Euro and Yen tranches. Using the
       exchange rates in effect on June&amp;#160;30, 2010, the facility had
       an outstanding balance of approximately $413.2&amp;#160;million in
       U.S.&amp;#160;dollars, which bore a weighted average interest rate
       of 3.9% and matures in October 2012.
   &lt;/div&gt;
   &lt;div style="margin-top: 12pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 2%; margin-right: 0%; font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: #ffffff"&gt;
       &lt;b&gt;&lt;i&gt;&lt;font style="font-family: 'Times New Roman', Times"&gt;Unsecured
       Credit Facility Guarantees&lt;/font&gt;&lt;/i&gt;&lt;/b&gt;
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company is a guarantor of the Operating
       Partnership&amp;#8217;s obligations under its $550.0&amp;#160;million
       (includes Euros, Yen, British pounds sterling or
       U.S.&amp;#160;dollar denominated borrowings) unsecured revolving
       credit facility that matures in June 2011.
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company and the Operating Partnership guarantee the
       obligations of AMB Japan Finance Y.K., a subsidiary of the
       Operating Partnership, under a Yen-denominated unsecured
       revolving credit facility, as well as the obligations of any
       other entity in which the Operating Partnership directly or
       indirectly owns an ownership interest and which is selected from
       time to time to be a borrower under and pursuant to the credit
       agreement. This credit facility has an initial borrowing limit
       of 55.0&amp;#160;billion Yen, which, using the exchange rate in
       effect on June&amp;#160;30, 2010, equaled approximately
       $621.9&amp;#160;million U.S.&amp;#160;dollars and bore a weighted
       average interest rate of 0.65%, and matures in June 2011.
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The Parent Company and the Operating Partnership guarantee the
       obligations for such subsidiaries and other entities controlled
       by the Operating Partnership that are selected by the Operating
       Partnership from time to time to be borrowers under and pursuant
       to a $500.0&amp;#160;million unsecured revolving credit facility.
       The Operating Partnership and certain of its wholly owned
       subsidiaries, each acting as a borrower, and the Parent Company
       and the Operating Partnership, as guarantors, entered into this
       credit facility, which has an option to further increase the
       facility to $750.0&amp;#160;million and to extend the maturity date
       to July 2012.
   &lt;/div&gt;
   &lt;!-- XBRL Pagebreak Begin --&gt;
   &lt;/div&gt;
   &lt;!-- END PAGE WIDTH --&gt;
   &lt;!-- PAGEBREAK --&gt;
   &lt;div style="margin-left: 0%"&gt;
   &lt;!-- BEGIN PAGE WIDTH --&gt;
   &lt;div align="center" style="margin-left: 0%; margin-right: 0%; font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: #ffffff"&gt;
   &lt;b&gt;
   &lt;font style="font-family: 'Times New Roman', Times"&gt;
   &lt;/font&gt;
   &lt;/b&gt;
   &lt;/div&gt;
   &lt;div style="margin-top: 0pt; font-size: 1pt"&gt;
   &lt;/div&gt;
   &lt;div align="center" style="margin-left: 0%; margin-right: 0%; font-size: 10pt; font-family: Arial, Helvetica; color: #000000; background: #ffffff"&gt;
   &lt;b&gt;
   &lt;font style="font-family: 'Times New Roman', Times"&gt;
   &lt;/font&gt;
   &lt;/b&gt;
   &lt;/div&gt;
   &lt;!-- XBRL Pagebreak End --&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       The credit agreements related to the above facilities contain
       limitations on the incurrence of liens and limitations on
       mergers or consolidations of the Parent Company.
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       If the Operating Partnership is unable to refinance or extend
       principal payments due at maturity or pay them with proceeds
       from other capital transactions, then its cash flow may be
       insufficient to pay its distributions to the Parent Company,
       which will have, as a result, insufficient funds to pay cash
       dividends to the Parent Company&amp;#8217;s stockholders.
       Furthermore, if prevailing interest rates or other factors at
       the time of refinancing (such as the reluctance of lenders to
       make commercial real estate loans) result in higher interest
       rates upon refinancing, then the Operating Partnership&amp;#8217;s
       interest expense relating to that refinanced indebtedness would
       increase. This increased interest expense of the Operating
       Partnership would adversely affect its ability to pay its
       distributions to the Parent Company, which will, in turn,
       adversely affect the Parent Company&amp;#8217;s ability to pay cash
       dividends to its stockholders and the market price of the Parent
       Company&amp;#8217;s stock.
   &lt;/div&gt;
   &lt;div style="margin-top: 6pt; font-size: 1pt"&gt;&amp;#160;
   &lt;/div&gt;
   &lt;div align="left" style="margin-left: 0%; margin-right: 0%; text-indent: 4%; font-size: 10pt; font-family: 'Times New Roman', Times; color: #000000; background: #ffffff"&gt;
       In the event that the Operating Partnership does not have
       sufficient cash available through its operations or under its
       lines of credit to continue operating its business as usual,
       including making its distributions to the Parent Company, it may
       need to find alternative ways to increase its liquidity. Such
       alternatives may include, without limitation, decreasing the
       Operating Partnership&amp;#8217;s cash distribution to the Parent
       Company and paying some of the Parent Company&amp;#8217;s dividends
       in stock rather than cash. In addition, the Parent Company may
       issue equity in public or private transactions whether or not
       with favorable pricing or on favorable terms and contribute the
       proceeds of such issuances to the Operating Partnership for a
       number of partnership units in the Operating Partnership equal
       to the number of shares of Parent Company stock issued in the
       applicable transaction.
   &lt;/div&gt;
   &lt;/div&gt;
</NonNumbericText>
          <NonNumericTextHeader>&lt;!--DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd" --&gt;
   &lt;!-- Begin Block Tagged Note</NonNumericTextHeader>
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>us-types:textBlockItemType</ElementDataType>
      <SimpleDataType>textblock</SimpleDataType>
      <ElementDefenition>Provides pertinent information about each guarantee obligation, or each group of similar guarantee obligations, including (a) the nature of the guarantee, including its term, how it arose, and the events or circumstances that would require the guarantor to perform under the guarantee; (b) the maximum potential amount of future payments (undiscounted) the guarantor could be required to make under the guarantee; (c) the current carrying amount of the liability, if any, for the guarantor's obligations under the guarantee; and (d) the nature of any recourse provisions under the guarantee, and any assets held either as collateral or by third parties, and any relevant related party disclosure. Excludes disclosures about product warranties.</ElementDefenition>
      <ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name Statement of Financial Accounting Standard (FAS)
 -Number 5
 -Paragraph 9, 10, 11, 12

Reference 2: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name FASB Interpretation (FIN)
 -Number 45
 -Paragraph 13, 16

</ElementReferences>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
  </Rows>
  <Footnotes />
  <NumberOfCols>1</NumberOfCols>
  <NumberOfRows>2</NumberOfRows>
  <HasScenarios>false</HasScenarios>
  <MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel>
  <SharesRoundingLevel>UnKnown</SharesRoundingLevel>
  <PerShareRoundingLevel>UnKnown</PerShareRoundingLevel>
  <HasPureData>false</HasPureData>
  <SharesShouldBeRounded>true</SharesShouldBeRounded>
</InstanceReport>
