Questions of this sort are often dependent on context. You don't mention where Paley and Seeger got that version, but it appears to be the set printed in Byron Arnold, Folksongs of Alabama, 1950, p 60 and reproduced in Bronson II, 70-71 (example 68.19). This was "Sung by Lena Hill, Lexington, Alabama, in 1945. Text written when a girl."

The bird sequence is often quite confused in American variants, and that seems to be the case here; some mis-hearings having crept in at some point. Whether or not Lena Hill understood any specific meaning by it is not indicated in Bronson, though perhaps the matter is mentioned in Arnold.

The verse usually contains offers, not menaces; and Hill's exact words appear to confirm that. Either Paley and Seeger altered one rather important little word, or you have mis-heard it.

Hush up, hush up, parrot she cried
Don't tell no news on me
All these costly beads around my neck
I'll apply them all to thee.