I'm not sure what the "or bird" part is here for. Birds are animals, seems kinda redundant... I always desire petting fancy mice. I guess since あなた put the bird part, I want to pet a chick. Chicks are just adorable. ....Also, Rowlet.
Well I 愛 pups but あなた really need to have a lot of time to get a dog for your pets especially when あなた are a student and your parents don't agree to get あなた one, so besides it I would 愛 to have a カメ and it seems my mom won't mind me having one.
Birds are classified as 動物 too though. Well, I would 愛 to pet an Eagle (Preferably a Golden one as it is also my お気に入り Animal in general). I have always admired their Grace and feeling of Freedom they inspire !!!!