I notice that in JY's novels, animals have some participation and supporting roles. Let's see if I can list some of them here:
GJ had 2 eagles and the horse which sweats blood (I forget its name)
YG or DGQB had that Divine Condor, although I am not sure the brother Condor can be categorized as a pet or not
OYF had snakes, lots of them (Indiana Jones will probably say, "Snakes. Why does it have to be snakes? I hate snakes." )
XLN had bees
Zhong Ling had marten
That Zhu family who tried to trick ZWJ in HSDS had a pack of dogs
ZWJ had that brother Ape (Gorilla?), but the same with YG's brother Condor that brother Ape might not be a pet. Also that... what do you call it, fire ice monkey (in the Fire Ice Island)?
Ding Chunqiu had lots of poisonous animals and insects
I do not remember anyone having cats, or bears. Anyway, feel free to add to the list.