Penguins are the most common animal on the Iceberg, an island that can be accessed by either purchasing an iceberg key for 12,500 coins or crafting it with 45 sapphires and a coconut. It is one of the most cost effective tier 2 pets, as fish are much easier to grow than the typical tier 2 foods.
Obtained through a quest given by the penguin queen.
You can find her mate at the base of the jungle waterfall. After speaking with her, return to the penguin queen to unlock the penguin egg cage.