From the only news source that matters, I found this article that says this scholar found a 1,600 year old Greek version told by a comedy duo of the day.  Only it wasn’t a parrot.  It was a slave.

The joke was discovered in a collection of 265 jokes called Philogelos: The Laugh Addict, which dates from the fourth century AD.

Hierocles had gone to meet his maker, and Philagrius had certainly ceased to be, long before John Cleese and Michael Palin reinvented the yarn in 1969.