I'm a hobbyist herbalist and I'm interested in this topic, too. There's herbs parrots can have, and there's herbs parrots can't have. Regarding many herbs, I'd leave medicinal dosages to an expert. You don't want the risk. Herbs heal but some, or too much, can be dangerous.
Asparagus and Sweet Basil is safe and very good for parrots; I feed both to Tiki all the time. Give the Sweet Basil fresh from your garden, or buy organic. Lightly steam the Asparagus. Dandelion greens are also excellent. Make sure it's really Dandelion, though, as Dandelion resembles other plants that could be deadly for a bird. There's no central stalk and the leaves aren't fuzzy with Dandelion. I know parrots can have cinnamon and it's good for them but I'd give it in very limited amounts. Cilantro is okay, too. I've given it to Tiki fresh.
Herbs to AVOID for parrots includes but isn't limited to: Comfrey, Licorice, Nutmeg, Poppy, Marijuana
I am not sure about Thyme, Mint, and Dill but I would ask a vet about those first. Personally I would stick to feeding the parrot known healthy bird herbs such as Asparagus. I can't help but wonder if Asparagus makes parrot's urine stinky, too, as it does for humans
