I think there are two different issues to consider. The risk of disease is one and the less serious one. You can use common sense and educate yourself to prevent that being much of an issue. The more serious risk is your legal situation and potential loss of your parrots if there's an outbreak of a disease in your area. Although having chickens on your property could increase the risk, just living near other places that have chickens, especially large operations, potentially puts your parrots in harms way. Some areas have never had a problem and some areas have. It makes a difference where you live.
There were a lot of horror stories coming out of CA in the past, during a disease outbreak, where parrots were confiscated and killed in front of their owners. It was bad. Really bad. Healthy looking indoor parrots, that weren't even tested. They lived in the eradication zone. Because of the severity and scope of the problem, they had a lot of temporary employees going door to door and some of them didn't handle things very well.
They had different rules back then. I think they are doing quarantines and testing of people's parrots now, but I never heard all the finalized details. I just know there was a lot of outrage over what happened and they tried to change the rules to make it better the next time.
I have no idea what state you live in or what their rules are in an outbreak. You should contact the government officials responsible for these situations in your state and ask them for information.
As far as managing actual disease risk, my parrot was bred in captivity and I had him tested for everything they could test for, when I got him. I've gotten my chickens as chicks from well known hatcheries and get them vaccinated for whatever the hatchery offers for vaccinations. I feed both species as healthy a diet as I can, so they will have healthy immune systems.
I have yard shoes that I don't wear in the house. I wash my hands after being in the coop or petting the chickens. I always wash my hands before handling my parrot, no matter what I've been doing or not doing. I have dogs, cats and fish, so I'm always careful about washing my hands, since parrots don't always handle the bacteria from other species all that well. After cleaning the coop, I take a shower and put on clean clothes. I also do that if I've been to a store with birds or visiting other's birds. I don't usually go to auctions or swaps, but if I did, I'd shower and change afterwards.
I've had small parrots since at least the early seventies and chickens on and off since shortly after that, without it being a problem. My CAG is in his early twenties these days.