There seems to be some confusion of ballads here (maybe the confusion is Pete's; he isn't above mixing sources)...

"False Knight on the Road," Child #3, is the song that begins:

The knight met a child on the road,
Where are you going to, said the knight on the road,
I'm going to my school, says the child on the road... etc.

The "parrot part" pretty much belongs to the ballad variously known as Henry Lee, Salt Water Sea etc. ... it's Child #4, Lady Isabel and the Elf Knight originally, with words to the effect:

Hold your tongue, pretty poll parrot, and say no word of me,
Your clothes shall be of the finest gold, no need for poverty

And as Nerd remarks, the opening really is mostly heard in the Bailiff's Daughter of Islington, though it's probably older:

There was a youth, and a well beloved youth, and he was a squire's son,
He loved the bailiff's daughter dear, that lived in Islington.

Untangling all these rubber bands is a job for Hercules (Child #749), and I haven't heard the Seeger song in question, but from the various descriptions above it sounds more like a medley than a distinct song in its own right.