I'm popping over from the Parrot Chat thread. I have two cats and two birds. One of my cats I have always trusted around birds. Strange as it is to say, she basically doesn't have a prey drive. Watching birds, small animals, etc, has never interested her. She's never even thought about going after my birds.
She is allowed in the room with my birds with them out, and even on the bed with them when they're on the bed, etc. Of course I still don't leave them alone out together, for safety, though even if I did I wouldn't be worried. This particular cat is also 20 now, so she's not really into playing with anything anymore anyway. My one cockatiel that does come out (the other is cagebound, which we are working on) has learned that basically this cat is not a threat. She does step away if the cat moves too suddenly for her tastes, and will hiss at the cat if she feels she's getting too close, but that's it.
My other cat is NOT allowed near the birds. She does eye them up. Even though she isn't that way with baby chicks and ducklings, the cockatiels are enough like wild birds, which she absolutely would kill if she got a chance, that it excites her interest. I know if she was left alone with them, she'd be having a birdie snack. So she isn't allowed in the room when the bird is out. I also don't clip mine, which means she could take off and land on the floor or somewhere else the cat could get to her before I could. Nope.
The birds aren't stressed out by the presence of this cat, but I also don't let the cat linger near their cages and when I'm not home, the cat is locked out of the room where the cages are.
Basically, know your cat and take appropriate precautions. If she likes to go after wild birds, she will probably go after your parrot.