In the parrot world, respected avian trainers and parrot behaviorists have long challenged the notion that the concept of "dominance" or "alpha status" makes much sense with birds. Trying to dominate a prey animal is just a bad strategy. Parrots don't have the same social structures or psychologies as dogs, who are predators and pack animals -- that's been part of the argument about separating out bird training approaches from old-fashioned dog training strategies.
It turns out that dogs aren't like dogs either. At least what we think is true about dogs turns out to be pretty outdated thinking, as well. To be fair, even modern dog trainers use positive reinforcement training and don't try to train canines through intimidation and force anymore. And while many in the public are starting to warm to the idea of training based on positive reinforcement, most people still hold on to the idea that dogs do have a dominance-based mentality in their social interactions.
Watch this video from the PBS program "Through a Dog's Eyes" to learn where the dominance myth comes from: