A parrot is a lot lighter than a cat. You roll over on a cat, and they will scramble out of the way as fast as they can. You're also more conscious they are there. Due to mild allergies, I try to keep them out of my bedroom. That's not always possible, though. My first cat would want to play during the night. Just as i was falling asleep he would drop a toy in my face. Fortunatly, he knew how to fetch so I could just lay there and throw it until he got tired and quit. As he grew into adulthood he let me sleep more, but sometimes he would curl up along my stomach, on top of the covers. After I got him a companion, I would be pinned down by twenty pounds of cat; one curled in my stomach and one in the crook of my legs (except in the winter wnen the second one deserted me for the radiator.) After my first cat died his companion got the choice space along my stomach or back. But she would lay on top of the covers and then LEAN on me. In my sleep I would naturally move away from this pressure. I would wake up in the middle of the night to find I had been moved over onto half of the bed. Later, I adopted a 15 year old cat with hyperthyroid condition, which made her quite skinny. She liked to crawl under the covers for the warmth. About an hour later it would get stuffy, and she would crawl out, waking me up. Then, about 2am, she would want to lay down next to my head, IN MY FACE! That was the point when she would get shooed out of the bedroom! Currently, Chelsie sleeps with me because I have to keep her separate from Lilly. If she has just eaten, she will settle down along my back and sleep. But as soon as she thinks I'm awake or she gets hungry enough, she will lay ON TOP of me, usually with her face in mine. Or she will lay on the bed near my face and give me a bath. She gets repeatedly shoved to the foot of the bed until I'm ready to get up.
I am not necessarily a restless sleeper, but I do move around. I have never rolled over on top of a cat, probably because they lay on top of the covers, and because they move out of the way when they feel me start to move. But I have two slogan buttons that read "Cats take their half of the bed out of the middle" and "I want a cat with a snooze alarm!" They're both true!