Hey Mercy.
I think I may just have an answer to your issue.
As Michael says, cats instinctively react to specific movements in a certain way. This is hardwired into them for rodents, but not for birds.
This is because predatory birds in the wild sometimes preyed on small wild cats and the same instincts that tell them to hunt some birds tells them to be wary of others. Dogs also carry a slight wariness with regard to predatory birds, but not to the same extent. The trick is to teach them that the parrot may be a potential threat so that they respect it, and reward them for ignoring or moving away from it once that is established. We need to make "Parrot" fall into the category of "Birds to respect and be wary of" in the mind of your cat.
My old budgie, Milo, who was eight years old at the time, died a month after a friend of my fathers surprised us with a purebred golden retriever puppy. The puppy couldn't help himself. They are bred to hunt birds, after all. My mother informed me that I could under no circumstances get another bird of any sort unless I could guarantee the dog would not eat him or her. It took me a year to think of this solution.
Using a dummy bird (a block of Styrofoam shaped into a head and body with feathers jabbed into it in the shape of wings and tail, slightly larger than a budgie and rubbed all over the bird cages at the pet store for parrot scent) I first used negative reinforcement to train the dog away from this "Parrot". I hung it from the ceiling fan from a thread, turned the fan on low, and got out my trumpet.
Then I opened the door, letting the dog into the room, and hid behind the couch.
Any approach of the "parrot" earned the dog a horrible noise. It only took two or three times to get him the idea, after which I switched to positive reinforcement: Taking the thread from which the "parrot" dangled in hand and slowly following him around the house with it. His moving away from the "parrot" immediately on approach was rewarded with a treat. I kept this up for about three weeks to be safe, though he had it down by the end of the first training session, and repeated it many times a day during those weeks.
I have had Peeper for five years now and Vivi for 13 months. To this day, Cooper turns around and slowly walks away if either one comes within five feet of him.
I believe this method could work fairly reliably for most indoor cats too, but not likely for outdoor cats, as 2/3 of outdoor pet cats actively hunt and kill birds and have already reinforced that instinctive behavior towards them. Just make sure that whatever you use for negative reinforcement can't be directly traced to you, so that the cat/dog thinks the threatening sound etc is coming from the bird dummy. It should also be a unique noise that you haven't used for training purposes before, so the animal associates the noise only with the approach of the parrot.
Make sure that you reinforce the "Bird avoidance" behavior really well and for an extended period, making sure all the cats understand it to the point of reflex before you consider bringing home a bird, and continue to exercise caution in their presence even after you've had the "real" parrot home for a good while.

A good test is to hang the dummy bird from a ceiling fan again and hide. If the cats avoid the bird like the plague even without your presence, consider this a success.
