I mean, it really depends. Horses aren't predators / carnivores and there's some examples of symbiotic relationships between larger animals like horses and small birds (i.e bison and cowbird) but horses can still be spooked easily and they ARE big while birds are small and relatively fragile.
As for dogs I'd just advise keeping the bird away from it. I'm not sure what kind of dog it is but a lot of common dog breeds have been bred to hunt and in most cases hunt birds, so you can see where that may go wrong even if your dog(s) haven't been trained to hunt birds it's still in their genetics.
Some people have no problem with their bird and interspecies friendships, but it's just a rather risky thing that's better left avoided.
A parrot and horse have no significant natural interactions or competitions. Obviously, the size differential mandates caution against an accident.
I have and have had many dogs that generally ignore my birds. In the distant past I learned a horrible lesson that proved to me one can never trust the two together.