You can not only breed to get pets but you can also create your own specific pet you'd like!
You need to go to the Adoption center (Pet care->adopt (https://legacy.atrocity-pets.com/adopt))
then you choose the Create! option
In this case we're gonna try to create a Catari
So once you click create you'll find a new menu
After you hit "Preview pet" You'll then get the choice to save/adopt the pet
Congrats on your new pet!!!