In my mind it was Telstar, by the Tornados, 1962. But I'm not 100% sure, I have many 45s that go back before then but I think I inherited most those from the parents. Lots of doo-wop, Elvis, and ballads - Anka, Cole, Bobby Vinton etc.
My parents purchased a record player from JJ Allens and I selected Frank Ifield's "I remember you" as our first record. My tastes changed dramatically after that!