Yes — a small amount of plain, fully cooked, unseasoned chicken is safe and a good protein boost. Yes, it feels odd that a bird eats bird; nutritionally it's fine.
Keeper's noteA few shreds of plain cooked breast, offered occasionally, is fine. A clean cooked drumstick bone also gives big parrots something to gnaw (supervised). Never fried, breaded, or seasoned chicken.
What to watch out for with chicken
Parrots are not strict vegetarians — wild birds eat insects and the occasional animal protein, and cooked chicken is a complete, easily digested protein source, useful during molt or recovery. The rules are about preparation: fully cooked (no raw meat — salmonella risk), plain (no salt, oil, onion, or garlic), and small portions, since it's rich.
How to serve it
Plain, fully cooked, unseasoned — boiled, baked, or grilled.