I don't think dominance is necessary. Now, I communicate with my Grey in certain ways she communicates with me (she "rebukes" me with a hiss sound... usually for not paying enough attention to her while she's on my shoulder, I've read this can be a Grey thing) and I will do the hiss sound back. But biting is something I don't want to encourage, and biting back -- strange as it may sound -- I think could encourage more biting.
But our parrots are companion parrots, and is dominance really the way to go? I agree with Marie that there's other ways.