With that answer it certainly seems like it would be her dad, and with her reaction it seems like she knows something about that, but why would she know something about that?
It doesn't look like he raised her much before he left, definitely not enough to show her he can turn normal cats into bakaneko unless that is something he does often. In which case I wouldn't expect Tiggy to be such a rarity. And it doesn't feel like something the mom would tell her without reason, and there is a good chance the mom wouldn't even know about it.