Welcome! We have a housecat and a parrot. Some cats have very high prey drive and it is inevitable that the motion/sounds of a parrot will trigger some kind of response from even the most languid of felines! Ours is a very spunky, active cat who will go into “hunt” mode even for birds outside the window, and as such she is not allowed anywhere near our parrot or his things, and we have multiple points of security (child locks for door, web cam on his room).
The reality is the risk of injury and death to your parrot is very significant. I am an anxious person by nature so I am disinclined to leave my parrot unattended if there is a possibility the cat can even gain access to the same room as him while caged; your mileage may vary.
Conveniently however, cats are crepuscular; so they tend to want to play first thing in the morning (which is when we give our bird his breakfast) & in the evening when your parrot will probably be starting to think about his bed. We find that has made it fairly easy to ensure both get their share of attention; that, and parrots (Amazons especially) need 12 hours of sleep, so while your birdie may go to bed at 8 or 9pm, your kitty-cat has several more hours to get all your attention!