Yes — fresh oregano is a safe, aromatic herb, fine for parrots in normal fresh-leaf amounts.
Keeper's noteFresh sprigs to shred and nibble, or chopped into chop. Dried plain oregano (no salt blends) works too. Never oregano essential oil.
Why oregano is good for parrots
Fresh oregano is non-toxic and rich in antioxidants, and is a common addition to herb-based parrot foraging. Offer fresh leaves in small amounts; as always, avoid concentrated oregano oil, which is extremely potent and not appropriate for birds.
How to serve it
Fresh sprigs, washed; whole or chopped.
Plain dried oregano is fine; avoid seasoning blends (salt).