The parents are completely unreasonable. So I'm wondering if neither have proof that the cat is theirs, could she possible just take the cat?
Ive been in a lawsuit with ex business colleagues. From that I learned that, just as Heidie says, the legal system isn't interested in word against word. In cases like that they'll pretty much say, though somewhat politer: "We're not interested in your boring bickering, all we want to see is the receipt, thank you very much."
If your mom won't contest that you bought it, Im pretty sure they'll say the cat is yours case closed.
I also once phoned a lawyer with a problem once for a no charge query whether I had a case, he gave me a rough explanation as to why I did and gave me a little too much info, which I used successfully. His voice was a little chilly when I phoned him and cancelled our appointment saying things had worked out...he realized he'd helped me too much on the phone.
A legal problem as small as this, you could probably just phone the local police station and ask (which will also look good in your papers, should the unlikelyhood of this going to lawyers occur).