"The origin of this rather odd term goes back to World War II and a radar beacon system called IFF (Identification, Friend or Foe). Allied aircraft were equipped with transmitters that replied to radar sweeps with a sound similar to a parrot's squawk."
I don't know anything about WWII systems making a sound like a parrot, but the early transponder system repeated information from a radar sweep just like a parrot would mimic something you said. The codename of the system was "Parrot" because it acted like one -- I hadn't heard that it sounded like one.
We still do "parrot checks" in the fighter world when referring to checking transponder/IFF codes and replies.