You can increase the quantity of Pet Columns by using  "Additional Pet Slot". When you reach Level 20, you can get one by Novice Reward. To make things better, find Vivian Chung in Prosper City when you reach Level 40 and find Lincy Year in the Dragon City when you reach Level 70 to accept 2 books.

If you want more pet columns, go directly to the store to buy "Additional Pet Slot".