How do you know your dad isn’t lying about receiving a Bachelors Degree? He might have lied for his honor, or that it would be disingenous for him to demand you get a BA if he never had one, so he lied about it. That could be the reason he is asking you to not put down that he has a BA. If he refuses to tell you where he got it, and your mom isn’t any help, I would think there is a reason he won’t disclose and you can just leave it off.
@Alfonsia, yes there are, particularly now with qualifications standardizing, but it has to do with the difference in systems and how int’l student offices assess transcripts from various countries. My husband ran into that, which is why he had to do two BAs. (Son reported both on the apps he filed. But I can see how OP’s dad might feel it wasn’t a BA if he was from a similar system. Just giving him the benefit of the doubt.)
@N’s Mom, that advice, “our history follows us, especially in the internet age,” should be emblazoned in big letters in every counselor’s office!!!