I like cats also. Got 7 at present, as momma cat had kittens 4 months ago. Had 3 birds but the last batch of kittens ate the birds, so we gave up on birds right now.
She will quite often run her beak through the cat's fur behind the ears etc. and the cat will relax and enjoy it for about a half-hour or so at a time.
They're good buddies, but their friendship is pretty much based on the fact that the parrot has a beak that can crunch a broomstick into splinters.
161 more and you'll be ordered to get rid of them like the women did last week. There is a thread about that somewhere here on FR.