I would say it came from misguided beliefs on how wolves act.....
The alpha male/female thing came from viewing unrelated wolves stuck together in an unnatural setting.... a wolf pack is a *family*. It's not about an alpha dog that is the strongest is thus the leader, but rather the *parents* of the pack are the leaders. Occasionally groups of families may pack together, or a lone wolf may join a family, but it's just that, a family. The younger members respect the older ones.
There's quite a bit out there about debunking the "alpha" myth!
De-Bunking the "Alpha Dog" Theory - Whole Dog Journal Article
Pack Theory Debunked | Victoria Stilwell Positively
Dog Training: Animal Experts Debunk the Alpha-Dog Myth - TIME
So I'd say yes, that belief that parrots are "dominant" creatures came from misbehaving dogs and they just slapped it onto parrots, too! Thankfully, positive reinforcement training (often under the guise of clicker training) is more about learning how to communicate with our pets through a positive manner while also enriching their lives!