Parrot and ape can be used as verbs, and are in dictionaries. We do not call a person a parrot typically, instead using it as a verb.
gerund or present participle: parroting
"encouraging students to parrot back information"
Essentially, it is a form of copycat, but copycat is a noun.
imitate the behavior or manner of (someone or something), especially in an absurd or unthinking way.
"new architecture can respect the old without aping its style"
As an insult, used as noun, "You big ape" is someone who is aggressive, large, uncontrolled
Monkey is also used as a verb
1 : to act in a grotesque or mischievous manner.
2a : fool, trifle —often used with around he likes to monkey around with engines.
Calling a child a monkey is sometimes as a joke.
They climb, get into trouble, act like a little monkey
We do use animals in insult.
Eats like a pig, or the opposite eats like a bird
Acts like a pig
is a snake in the grass
Trump has called women "dogs" (He dislikes dogs, first US President in a long time without one).
blind as a bat (Bats are not blind actually)
Many people consider imitating someone to be a sign of admiring them.
I can say English has similar meanings, but as a noun is different and usually used as verb for monkey and parrot.