We dogsit on rover.com, and have had - on the high end - 5-6 dogs in the house (2 clients only at a time, three if the third one is our neighbor, a sweet/docile/low maintenance mini poodle).
As stated by my friends above, same household? Sure! Intergrated? No.
You may get to a point where you are fully comfortable - say, while watching TV - with dog sleeping next to you while you hold the bird, or maybe the bird is on a stand next to you (with you between the two).
the key is always supervised, always. And always you between the dog and the bird. Or the bird be completely untouchable, like on the top of a door. Even then, only if the bird is calm and not trying to fly around.
Always supervised, always vigilant.