Little Poll Parrot
Sat in his garret
Eating toast and tea;
A little brown mouse
Jumped into the house,
And stole it all away.

This is no doubt the source of the use of the word parrot as a verb
(transitive), meaning repeat without really understanding. As in
“They parroted my parrot jokes, but none of them laughed.”

Polly want a cracker? The stereotypical parrot sentence,
whether said to a parrot, or by a parrot.
Possibly popularized in Robert Lewis Stevenson’s Treasure Island