Sapins

If you'd purchased the iMac from an Apple retail shop or online shop then on their official receipt there is a part number alongside the serial number in the form of XYYYYX/X X=>Letter, Y=>number.

Not sure about the Amazon receipt format but if you don't see part number then try the order/marketing number if there is one.

After all Amazon should sort this out for you, I suppose.

Sapins

Not sure why Apple'd want a Part Number from you, please follow the below procedures if you did differently & see if it helps.

1) On the Up To Date form, fill in the Date of Purchase =>Place Of Purchase =>select Reseller =>Details of Reseller (Amazon,in your case) => check I accept Terms & Conditions =>Click Continue. Leave Right to Copy & Number of Qualifying systems for right to copy options blank.

2) Enter your details =>Enter your contact informations (just enter 44 without the + sign in the Country Codes if asked)=>Click Continue =>Enter Serial Numbers(no Part numbers is needed)=> select system (iMac in your case) => Click Continue & submit your form ,you should get approval from Apple at once & an email will be sent to your email address with a file to download & a Password to open it.

3) You'll get a consent code within the PDF file which you can use to redeem Mountain Lion in App store, anyway if Apple still fails to approve your upgrade then only if you got the iMac from a marketplace seller because Amazon is an approved Apple product seller, they're supposed to help you out.