I don't know. "On tape that changed my life" sounds almost as though the material tape, and not what was recorded on it, had this life-altering effect.
I assume what actually changed your life was hearing the Beatles' debut. Of course it might well be the fact of your father buying you something. It's hard to say.
If it is the hearing, then that life-change is not part of the presentation of the tape, and so I'd suggest a future tense:
My father bought me a tape of the Beatles' debut, which would change my life.
(You could as well use "a copy on tape of" if you prefer.)