There’s a lady by the name of Dr. Kathleen Walker-Meikle who’s written three books on medieval animals: Medieval Dogs, Medieval Cats, and Medieval Pets. They’re all fairly short books, but they have pictures and huge numbers of footnotes. They include a lot of useful info as well as human interest, so they seem worth a browse. (And if you’re looking for Christmas presents, Medieval Cats and Medieval Dogs are both priced very reasonably; and Medieval Pets isn’t much more expensive.)
Anyway, it turns out that there’s a reason why parrots appear in so many pictures of Mary and the Child Jesus, and why so many Annunciation St. Gabriels and other angels have parrot-colored wings.
There was a legend (probably derived from parrot training methods of the Classical period and the Middle Ages) that every parrot on earth just naturally knew how to say, “Ave!” which is to say, “Hello!” Now, it’s a little doubtful that Nature caused this; but it is a charming connection between parrots and St. Gabriel! (And it’s possible that people interpreted natural parrot sounds that sounded sort of like “Ave” as being just a mangled attempt, which would lead to them saying “Ave” to the parrot and praising versions of the noise that seemed more like “Ave.”)
Here’s a page of Marian and Christological parrots from the University of Dayton’s Marian Library. It notes that parrots can also represent words and thus the Word.
I’m having trouble finding more of these with search terms, but if you’ll look, there are a fair number of green and red winged Gabriels out there, and a fair number of Christ Childs spending time with parrots.