I used to rehabilitate parrots and I also rescued pigeons - I was always told by the vets I worked with that parrots are highly susceptible to the things pigeons (and other wild birds and even other parrots) can be carriers of.
Not necessarily diseases, but things like coccidia, canker, worms, e coli, salmonella, and avian-specific diseases can be transmitted to other birds through various means: air, touching, lice, your clothing, your hands, etc. These kinds of things can lay dormant or be maintained in low numbers, but are generally shed or surfacing during stress - which rescued pigeons undoubtedly find themselves under while getting used to their new captive environment.
I'm in no way saying pigeons are "disease carriers", but like any wild animal they are hosts to their own natural assortment of bacteria, parasites, single celled organisms, and
potentially viruses. Captive parrots on the other hand are generally
under-exposed to such things, and can react very badly and succumb quickly when they come in contact with these things suddenly.
Use great care when housing pigeons and parrots under the same roof. I would tend to the parrots first (including feeding and watering) after thorough hand washing, and with clean clothing. Then deal with any rescued feral pigeons. Only return to the parrots with clean hands and clean clothing. I would not let the parrots traverse anywhere in your home where the pigeons are.
Best,
Kari Jo