Q: Why was the pirate sad when his parrot left?
A: It gave him the cold shoulder!

Explanation: Yes, the last parrot joke for a while.

Have you seen those pictures of a pirate with a parrot sitting on his shoulder?  Take a look at this one, if you haven’t.  (That link will also explain about the history of parrots and pirates.) I have never had a parrot sit on my shoulder, but I would expect the parrot to feel warm on my shoulder.  Then, when the parrot flies away, my shoulder would feel cold.

“To give someone the cold shoulder” means to purposely ignore that person

This joke is funny because it plays with the phrase cold shoulder, in both the literal sense of feeling colder, and in the meaning of the idiom, to ignore someone.  Perhaps the parrot was ignoring the pirate, giving him the cold shoulder?

Here is Adele with her song Cold Shoulder–