We need to be careful about the nuances of different languages here.
1. repeat after me
This means "Say again what I have just said in the way I said it in order to learn it". It is a conscious instruction.
In Chinese, it is "我每说一句，你照样跟着说一句，把这个学会。"
This applies to singing as well. The verb is 唱. Replace 说 with 唱.
2. He just repeats what I say, like a parrot.
3."Stop repeating everything I say."
This is not neutral at all, just as the English original is not neutral. The English original is quite rude and peremptory (unless when you are playing with a close friend or a lover).