Parrotlets are of a similar body size, but different birds in reality.
They have broken skin before. Never because we sought it out. Neptune was in a fight with Picard. He was bleeding it was very dramatic. I had to seperate them. This was the worse bite I have gotten. Breaking the skin is relative. I don't have any scare. They are tiny and can't really harm us. It isn't like they can take a finger. It is like being pinched. Picard gets protective of my partner. Picard has nipped at my feet and it was very pinchy. He wanted Eric to himself!
I think the bigger issue with a small bird is you need to control yourself when bit. If I reacted when Picard nipped my feet I could have easily stepped on him. He kept charging for more. I had to sit on the couch and lift my feet out of his aggro range. Or if one bites your finger and pull away you could hurt them.
It can happen, but dont worry. Honestly if you respect the boundaries and read the body language bites are rare. They let you know when they are not having it.
Hope it helps.