I'm training my pets (apparently they're both on a battling kick because of the plot), and I noticed at the training center, Jim says he can't train pets that are over level 5000. I'm wondering: how would you train pets over level 5000?

You could always use stat boosters.

Stat boosters are the only way I can think of as well. The highest level pet is currently in the low 3000s. They might even implement another training method when 5000 is reached, idk

Ooh okay. I should have known, lol :P

You could also use exp, I guess.