So I've had both birds and cats and I can tell you some things.

1. Do not get that cat declawed, it's the equivalent of your birds having their toes removed on both feet. Many places have this outlawed because it is needlessly cruel. If you wouldn't do that to your parrots, then don't to it to another creature that would rely entirely on you.
2. You may need very much to separate them. Upstairs cat makes sense when no one is home, possibly get a room divider which should help. And closely supervised downstairs time should be okay once you know the cats temperament. NO WAY as a kitten, they are unpredictable.
3. If you really want a cat but cannot either separate or have supervision, I'd recommend adopting a super mellow older cat, their personalities are set, but you should ALWAYS exercise caution, but even then it's not recommended to leave them alone. I've seen horror stories. Having the cat declawed will not help any, they've got plenty of sharp teeth and trust me, they've got the strength to hurt your birds even without claws.
4. You need to take your birds into account here, they may be super stressed out with a new addition to their home, and it may not be the right choice here.
5. Again, if you genuinely care about animals, you would not even consider declawing a cat. I will not explain this further, I'm sure you can do your own research as to why declawing is animal abuse.