"mimic like a _____ (animal or something else)"?

In my mother tongue we use monkey to describe a person who imitates or mimics a certain behaviour much like a copycat in English. I am surprised that the same image is not used in English, and even more surprised a cat was used to express this meaning. I don’t see how a cat imitates things, it is actually known to be a rather self-willed animal that knows how to get what it wants, so to say. Probably that’s why you have to add "copy" at the beginning, whereas monkey has the meaning of imitation inherent in it, a bit like parrot (with the difference that "parrot" refers to imitating words somebody said or the way they speak.)

So if I see someone imitating me, I can say in my mother tongue:

Don’t be a monkey! (Which in English would be Don’t be a copycat)

Are there any idioms or words that would express the meaning of imitating someone’s gesture or behaviour? It can include the name of an animal or not. The sentence could look something like the one I gave already:

Don’t (be (such) a) ____ !

or

What are you? A ____ ?

Note that my sentences are similar to what I would say in my mother tongue. However, the English idiom might have a different structure, so other ways of saying this would be acceptable.