That can't be true. Invoice is the proof of purchase and is permissible in a court of law in case something goes wrong and you want to bring the culprits to consumer forum. I strongly suggest NOT to buy without getting an invoice.
This is what WD states in its warranty policy.
The term of your limited warranty period shall commence on the purchase date appearing on your purchase receipt from an authorized distributor or authorized reseller and extends only for the period of time set forth in the Product documentation.
Proof of purchase shall be required to be eligible for this warranty and to establish the commencement date of this warranty.
Link: *support.wdc.com/mobile/mpolicy.asp