In short, it is possible to catch a disease from a parrot, but it is
vanishingly rare. There is a disease called psittacosis which can be passed onto humans, obviously provided that a bird has it in the first place, and the bird would have to be very obviously sick and actively shedding lots of the virus for it to be contagious. I've just had my princess parrot treated for it, and I had a GCC some years ago who had it, and neither of them passed it on to me. In my almost 50 years of being around birds, some of whom may have had it years ago without my knowledge, I have never had so much as a sniffle. Upon my telling my doctor back in oh about 2014 or so that my GCC had it, he told me if I caught it he wanted to make me a case study because he'd never had a patient with psittacosis ever.
Another condition called "bird fancier's lung" can be caused by an allergic reaction to proteins in droppings or feathers, or even from feathers in bedding if you're VERY unlucky. But this condition is again pretty rare, and more commonly afflicts those who work with large numbers of birds, generally pigeon fanciers or occasionally pet shop employees who may be exposed to many birds, rather than just a domestic pet or two. Not unknown, but really not common in a setting like yours. It's good to be aware of these things so that you could advise your doctor in case you do happen to develop symptoms, but I honestly don't think you have a great deal to worry about