We have a 6 year old rescued greyhound, 6 year old shar-pei mix, 2 canaries, and our grey.
One time I had Kenzie (our TAG) on my arm and she was listening to the music of the canaries in the other room. So I took her in there because the canaries were in their cages and Kenzie can't fly very well. Before I knew it, she latched onto one of their cages! Luckily they were smart enough to stay close to the bottom. Kenzie was in the strangest trance and just getting her off of the cage was difficult because she wouldn't even listen to "step up!" She'd kill them if out together, so that's never gonna happen.
The dogs don't pay any attention to our grey. Even if she is out playing on the cage door, the dogs will walk right past her and not pay any attention to her. However, we don't let either of them get close enough because we don't want to take any chances...especially since we rescued a dog that was trained to chase things so she'd run faster!
I hope one day to get a Quaker Parrot, and HOPEFULLY they will get along in the future.