How we change what others think, feel, believe and do
Psittacism is parrot-like repetition.
Who's a clever person? Who's a clever person? You!
That's right, that's right, I'm sad and blue.
(other person says 'I want to go out') You want to go out. Good to hear. Let's go.
(other person says 'I'll be there in time') In time, you'll get whatever you want.
Parrots do two forms of repetition: repeating the same sentence, and copying what the other person says, word for word.
Repeating the same sentence is a form of emphasis, sending a signal that 'these words are important' and may be used for gaining attention to the words being repeated. Some people have a habit of regularly repeating the last few words you say, which can also be rather annoying.
Parroting as repeating back the words that the other person has said is a simple way of building rapport. This needs care as mimicking can be seen as an insult or crass and rather blunt method of attempted persuasion.
A subtle device is to use the last words that the other person said, but in a different way, almost as if you are picking up where they left off.