Just look at this massive list of Timeless Isle pets. Not surprisingly, a lot of them come from the rare spawns on the Timeless Isle, as well as other sources.
To collect the Celestial pets, you will need to complete the Celestial Tournament ten times.
There are three engineering pets: two are made while one is obtained from Blingtron.
Interact with the Neverending Spritewood (purple rare spawn crystal) and kill Scary Sprites (note these are the larger sprites which can randomly spawn as opposed to the smaller Angry Sprites) – (45.4, 73.3)
Ruby Droplet – Drop from Garnia
To reach Garnia, use the Timeless Isle Airways
Collect Bonkers by opening Timeless Chests inside Kukuru’s cave (59, 40) on the Timeless Isle. You will need to purchase a Kukuru’s Cache Key for each chest you wish to open. Each one has a small chance of containing the pet.
Purchased from Ku-Mo for 7500 Timeless coins