A lot of who the dominant parent is depends on the roles of the parents in the workforce. Women or men who stay at home are usually dominant simply because of the amount of hours they are tending to the children and making decisions for them. In a two parent working household these relationships may be more equal although I still believe it would be nearly impossibly to have it completely 50/50.
Yes, usually one parent does play a more dominant role because the other isn't around as much. Usually the other is working more, or at least, is working or gone at the times that the children are at home and awake. There also simply be one that is around more because the parents aren't together.
I think generally, one parent does take a more dominant role in raising a child. I think this is easily explained in that, in most families, at least one of the parents has a full time job. This leaves the other parent spending the most time caring for the children, giving them the more dominant role.
Some would argue that exposure dictates the impact a parent may have on a child's development. I would argue that there are many factors that lead to a dominant figure, but there is always a dominant figure. The mother has traditionally been the caretaker while the father has traditionally been the enforcer. When analyzing who had more impact, the question of "impact on what?" must be considered. In the traditional model, work ethic would be dictated by the father, and empathy would be more impacted by the mother. There is still always a dominant force.
There are plenty of families that both parents play similar roles in raising the child, I'm in one of them. There doesn't need to be one parent that is the assertive one and one that isn't, it's better if there is consistency between the two to establish boundaries with the child.