Yes, I should say hay comes in all qualities. Cows can literally eat anything. They can eat dusty hay, moldy hay, horrible black, rotten hay. Horses can not. If there is dust or any smell of mold, you need to toss it. So I would be less concerned about hay for the parrots if it has passed my horse inspection.
It also comes in different types of grasses. Some is like the grass that we grow on our lawn, some is alfalfa, some is clover and most is a mix between at least 2. How it is stored is important as well. To prevent mold, it can't get wet. And often barns have raccoons, rats, mice, birds and even owls who may be peeing and pooping in it.
So if you were to buy it for your birds it might be safest to buy right out of the field as soon as it's baled and store it yourself.
Storing hay isn't really fun though, lol! It takes up a lot of room and it's very messy. It has even been known to spontaneously combust if it does get moldy and then too hot. So I'd probably just stick with greens you can buy in the grocery store or grow in your own garden.