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 it depends on individual family upbringing. But if you ask me, I personally feel that anything within the house with just parents and siblings should be fine. As long as not totally naked. Not wearing bra at home is perfectly normal. You ask your mum does she wear a bra at home. If she don’t wear, means you also don’t wear. I don’t think it bothers your dad unless he himself is the pervert kind of guy. Usually fathers and brothers at home don’t really bother about whether the ladies in their house are wearing a bra.