My mom says that I'm not allowed to be braless under clothing, even if I'm in pajamas ready for bed, because my dad lives in the house. Is she being paranoid?

I think she is. It is your dad, after all. Unless she doesn't trust her own husband. If it makes your mom so uncomfortable, when you are not in your room, maybe throw over a cardigan or hoodie over your pajamas. That's what I do if I am at home and if my parents or in-laws (or pretty much any other guests) come to stay.