I would always be real careful housing anything with a pigeon! They're known to carry all kinds of parasites and diseases.
No, they're not. They carry no more diseases or parasites then your parrot, or the local song birds.
And the reason pigeons come back is an instinctual homing trait - so they wouldn't really work for a parrot, as a parrot comes back due to a call.
And unless you're housing budgies, Linnie's, etc. small small parrots with a big big pigeon (like a giant runt, Swiss mondaines, etc.) then the parrot will always be attacking the pigeon. I recently saw a friend who rescued a wood dove who was housed with a hook bill (not sure what type). Lets leave it at, it wasn't pretty. And pigeons aren't really big as much as they are thick, flying breeds (homers, rollers, tumblers, tipplers especially) have loads of muscle, which is why despite being fairly small, they have a lot of strength.
To end it, don't ever house a pigeon & parrot. They can interact (supervised, as all birds should be) but housing them together is a death sentence.
Sorry, I know this is kinda old, but wanted to clarify it wouldn't work and
why