Here is my code again with the updations you suggested :

Controller:
 

public with sharing class CustomInv {
    public Opportunity opp{get;set;}
    

    public List<npe01__OppPayment__c>  Rec {get; set;}
    public List<Contact> Contact {get; set;}
    public List<Contact> Contact1 {get; set;}
    public List<Contact> Contact2 {get; set;}
    public String oppId {get; set;}
    
    public CustomInv(){
        getOpp();
         oppId = ApexPages.currentPage().getParameters().get('id');
        System.debug('===============oppId:'+oppId);
        
          
        Contact = [SELECT Name, AccountId FROM Contact WHERE  AccountId=: opp.AccountId and (Title='Primary_M' or Title='Primary_D')  limit 1] ;
        Contact1 = [SELECT Name, AccountId FROM Contact WHERE  AccountId=: opp.AccountId and (Title='Primary_M')  limit 1] ;
        Contact2 = [SELECT Name, AccountId FROM Contact WHERE  AccountId=: opp.AccountId and (Title='Primary_D')  limit 1] ;
        Rec =[Select CreatedDate, SystemModstamp,npe01__Payment_Amount__c, npe01__Opportunity__c, npe01__Paid__c, npe01__Scheduled_Date__c FROM npe01__OppPayment__c WHERE npe01__Paid__c = true and  npe01__Opportunity__c = : oppId limit 200];
    }
    public Opportunity getOpp() {
        
        opp = [SELECT Id, AccountId, Account.BillingStreet, Account.BillingPostalCode, Account.BillingCity, Description, npe01__Payments_Made__c, Total_Amount__c, Balance_Remaining__c, Amount, (SELECT PricebookEntry.Name, Description, ListPrice, Quantity, SubTotal_Price__c FROM OpportunityLineItems), (SELECT Id, Name, Invoice_Date__c FROM Invoices__r) FROM Opportunity WHERE Id= :ApexPages.CurrentPage().getParameters().get('id')];
        if(opp.Description != null) {
            if(Pattern.matches('[A-Za-z]{3}[0-9]{3}[a-zA-Z0-9]{3}', opp.Description))
                return opp;
            else {
              ApexPages.Message myMsg = new ApexPages.Message(ApexPages.Severity.ERROR,'Error: Invalid Input.');
               ApexPages.addMessage(myMsg);
            }
        }
        return null;
    }
   
    Public PageReference save() {
        update opp;
        return Null;
    }
    
}

VF Component
 

<apex:component controller="CustomInv" access="global">
    <apex:attribute name="opporId" description="Attribute on the component." type="String" required="false" assignTo="{!oppId}"/> 
    <table border="0" cellspacing="0" cellpadding="0" width="100%" id="table1">
<tr>
    <td>
        <img src='{!URLFOR($Resource.Logo4567687)}' title="logo" />
        
    </td>
    <p align="center"><font face="Arial"><i>{!$Organization.Name}</i></font><br/>
    <font face="Arial"><i>{!$Organization.Street}</i></font><br/>
    <font face="Arial"><i>{!$Organization.City}</i></font>
    <font face="Arial"><i>,{!$Organization.State}</i></font>
    <font face="Arial"><i>{!$Organization.PostalCode}</i></font></p>
    <!--<td  align="right"><font face="Arial" >
    <b>Invoice for {!opp.Account.Name}</b></font><br/>
   </td>-->
   
  
    
</tr>

<hr/>

</table>
<table border="0" cellspacing="0" cellpadding="0" width="100%" id="table1">
<tr>    <td width="65%"><font face="Arial" >
        Federal Tax ID: 27-0580207<br/>
        </font></td> 
        <!--<td width="50%">&nbsp;</td>-->
   <td width="50%"><font face="Arial">Invoice number: <apex:repeat value="{!opp.Invoices__r}" var="line2">
   {!line2.name}</apex:repeat></font><br/>
  
   
   <font face="Arial">Invoice Date: <apex:repeat value="{!opp.Invoices__r}" var="line2">
   {!line2.Invoice_Date__c} </apex:repeat></font></td>
 
   
  
</tr>
</table>

<br/>
<hr/>
<p><b><font face="Arial" color="#000080">Account Information</font></b></p>

<table border="0" width="100%" id="table2">
<tr>
       <td colspan="3">
           <font face="Arial">Child name: {!opp.Account.Name} <br/><br/></font>
       </td>
</tr>
Parent Name:
<!--<apex:dataList value="{!Contact}" var="con">

<tr>
       <td>          
           <font face="Arial"> {!con.Name}<br/>
                             {!opp.Account.BillingStreet}<br/>
                             {!opp.Account.BillingPostalCode} {!opp.Account.BillingCity}
           </font>
        </td>
        
</tr> 
 </apex:dataList> -->
<!-- <apex:repeat value="{!Contact}" var="con">-->

<apex:repeat value="{!Contact1}" var="con1">

<apex:repeat value="{!Contact2}" var="con2">
<tr>
       <td colspan="3">          
           <font face="Arial"> {!IF(((opp.Description=='Mother')||(opp.Description=='mother')||(opp.Description=='Mom')||(opp.Description=='mom')),con1.Name,IF((opp.Description=='Father')||(opp.Description=='father')||(opp.Description=='Dad')||(opp.Description=='dad'),con2.Name, 'con.Name'))}<br/>
                            
                             {!opp.Account.BillingStreet}<br/>
                             {!opp.Account.BillingPostalCode} {!opp.Account.BillingCity}
           </font>
        </td>
        
</tr> 

</apex:repeat>
</apex:repeat>

</table>
<br/>
<hr/>
<p><b><font color="#000080" face="Arial">Invoice Detail</font></b></p>
<table border="0" width="100%" id="table4">
<tr>
    <td bgcolor="#C0C0C0"><font face="Arial">Item</font></td>
    <td bgcolor="#C0C0C0"><font face="Arial">Description</font></td>
    <td bgcolor="#C0C0C0"><font face="Arial">List Price</font></td>
    <td bgcolor="#C0C0C0"><font face="Arial">Qty</font></td>
    <td bgcolor="#C0C0C0"><font face="Arial">Total Price</font></td>
</tr>

<tr>
       <apex:repeat value="{!opp.opportunityLineItems}" var="line">
       
          <tr>
              <td>{!line.PricebookEntry.Name}</td>
              <td>{!line.Description}</td>
              <td>{!line.ListPrice}</td>
              <td>{!line.Quantity}</td>
              <td><apex:OutputField value="{!line.SubTotal_Price__c}"/></td>
          </tr>
       </apex:repeat>
</tr>

<tr>
        
       <td bgcolor="#C0C0C0" align="right" colspan="4">
       <font face="Arial"><b>Total:</b>&nbsp;<apex:OutputField value="{!opp.Total_Amount__c}"/></font></td>
</tr>

</table>
<table border="0" width="100%" id="table4">
<tr>
       <td bgcolor="#C0C0C0"><font face="Arial">Post Date</font></td> 
       <td bgcolor="#C0C0C0"><font face="Arial">Due Date</font></td>        
       <!--<td bgcolor="#C0C0C0"><font face="Arial">Description</font></td>  -->  
       <td bgcolor="#C0C0C0"><font face="Arial">Credit</font></td>
       <!--<td bgcolor="#C0C0C0"><font face="Arial">Debit</font></td>  -->    
</tr>
   <apex:variable value="{!1}" var="rowNum"/>
     <apex:repeat value="{!Rec}" var="Pay">
         <tr>
           
             <td> {!Pay.npe01__Scheduled_Date__c}
             <apex:variable var="rowNum" value="{!rowNum + 1}"  /></td>
             <td>{!Pay.npe01__Scheduled_Date__c}</td>             
             <!--<td>{!line.Description}</td> -->                       
             <td><apex:OutputField value="{!Pay.npe01__Payment_Amount__c}"/></td>
             <!--<apex:variable value="{!opp.Amount}" var="amt"/>
             <apex:variable value="{!Pay.npe01__Payment_Amount__c}" var="pmt"/>
             <td><apex:variable value="{!(amt-pmt)}" var="debit1"/>
             <apex:variable value="{!((-pmt*(rowNum-2))-pmt)}" var="debit2"/>
             
             $<apex:outputText value="{!debit1}"/></td>-->
             
          </tr>
        </apex:repeat>
 
      <tr>
        
       <td bgcolor="#C0C0C0" align="right" colspan="2">
       <font face="Arial"><b>Total Credit:</b>&nbsp;<apex:OutputField value="{!opp.npe01__Payments_Made__c}"/></font></td>
       <td bgcolor="#C0C0C0" align="right" colspan="4">
       <font face="Arial"><b>Balance:</b>&nbsp;<apex:OutputField value="{!opp.Balance_Remaining__c}"/></font></td>
</tr> 
         


</table>
<br/>
<hr/>

<p align="center"><font face="Arial"><i>Copyright {!$Organization.Name}.</i></font></p>
</apex:component>

vf email template:
 

<messaging:emailTemplate subject="Your requested invoice" recipientType="Contact"  
relatedToType="Opportunity">
<messaging:htmlEmailBody >
<table border="0" cellspacing="0" cellpadding="0" width="100%" id="table1">
<td>
Dear {!recipient.name},<br/>
        
        Following your order, please see attached and below your invoice per your request.<br/>
        
        Feel free to contact me if you have any questions.<br/>
        
        Regards,<br/>
        {!$User.FirstName} {!$User.LastName}<br/>
</td>
</table>
<br/>

<c:CustComp opporId="{relatedTo.Id}"/>




</messaging:htmlEmailBody>


</messaging:emailTemplate>