As others have mentioned, any dog you get will require training and no breed is guaranteed to be "bird safe", even if they are known for having low prey drive and a soft mouth.
I live in the country and we have free-range chickens, ducks, turkeys, and peacocks. Our dogs must be able to recognize that the birds are "off limits", so I consider prey drive and trainability when selecting new dogs.
One of our earlier dogs was a dachshund/terrier mix with a very high prey drive and stubborn temperament. The only thing that would stop him from killing chickens was a sturdy fence, preferably electrified. We were never able to train him out of this behavior. At the time, we also had a corgi and a standard poodle.
The poodle was very willing to please and easy to train. Poodles were originally hunting dogs and they can have a decent amount of prey drive. Standard poodles (the big ones) still retain many of those instincts, but they are so people oriented and willing to learn, it is not hard to train them to do whatever you need them to do. As long as you don't mind the grooming requirements, a standard poodle is a great dog - just keep in mind that you will need to teach them what you want.
Our corgi had a naturally low prey drive and never showed the slightest interest in chasing chickens, but she was an odd dog so I can't say if this is typical for the breed.
Our current dogs are a doberman, labrador retriever, and an Anatolian shepherd.
All of them have a decent amount of prey drive, which I have channeled towards rat-hunting. As puppies, they all had trouble with the chickens initially. It took time for the dogs to learn that I was not happy if they chased (or killed) our chickens. My doberman was the fastest learner, even though she has the highest prey drive of the three dogs.
The labrador is a pretty fast learner in general, but her prey drive is also quite high and she likes to chase. She eventually learned that I didn't want her to hurt the chickens, but she is not as obedient as the doberman or as reliable as the shepherd. She still enjoys running at the free range birds, but she will break off at the end instead of grabbing them. It is an annoying behavior, but at least none of the birds are harmed by her antics.
She looks so innocent when she is sleeping. Don't be fooled! She is a chicken-chaser.
The shepherd is a livestock guardian breed. He has the lowest prey drive, but he is also the least biddable and hardest to train. They are very independent minded dogs and it took a while for him to reach a maturity level where I felt he could be trusted around the birds. He basically had to decide for himself that he should leave them alone. It was a very slow process. And this was despite starting with exposure training when he was a very young puppy.
Personally, I suspect that the labrador was a bad influence on him. She taught him her "chase" game. And it took a while to get him back on the right path.
...
I wouldn't let any of my dogs into the aviary with my parrots. It might be possible to teach them to leave the birds alone, but the cost of failure would be too high. Regardless of what breed you end up getting, I would recommend relying on separation techniques more than low prey drive to keep your birds safe.
And keep in mind that that the personality of the individual dog as going to be just as important as the breed. Ideally, you will want a dog that is people-oriented and eager to please, like a poodle or doberman, as that desire to make you happy can help overcome the dog's natural instinct to kill.
But definitely avoid terriers. They are fluffy little murder machines.