Here is the full list of all 15 The Curious Tale of the Stolen Pets achievements.
Open all doors in all worlds
Pick up all unique objects from the hat
Throw a snowball at the train in the Winter Vacation World
Find all the animals in the winter vacation world
Find all the animals in the aquarium world
Find all the animals in the prehistoric world
Find all the animals in the magic world
Create pearls of all rocks in a single round
Let the meat eating plant eat watermelon
Collect all the coins
Get the true ending by finding all the animals in all worlds
Complete the story
Pick all the berries in all levels
Find all the animals at the summer house
Poke the swing while dog is on it