Would his All Starr Band be his All Parr Band?
I mentioned that earlier
This is faulty logic on Ringo's part. If "Starkey" is on his birth certificate, then it is indeed his "real" name. Parkin would be his family name or what have you. It's like Marilyn Monroe- even though she was born Norma Jeane, her "real name" at death would be Marilyn Monroe because she changed it legally.
When your name is changed legally, you get a new birth certificate with that name on it.