Traditionally, father was the head of the household, but by the rules of manliness, no male would ever get much involved with the task of raising a young child any more than he would involve himself in ordering the kitchen - that was women's work and women's realm, back then. For a big strong man to meddle in such matters would be the equivalent of sexual perversion or profound effeminacy, and was strictly taboo. Only the most general influence of the male over such matters would be condoned by society. It's terrific that we've now altered some of the ways in which women were discriminated against in the past, but it's not helpful to go on to infantalize our view of women in history: they did have some power all of their own, even then.
Nowadays, parental decision making is more complex and diffuse, and there's usually no "third vote" of society or the church to decide matters in case two parents disagree. What to do?
Well, for a start, modern psychologists strongly believe that hiding differences in parental opinions from children is harmful, and messes up their thinking; that it's healthier for the child to know who thinks what - the child obviously shouldn't be in the position of being the deciding vote, of course, just that the child should be aware what was behind decisions and how they were made, rather than have life be an unpredictable series of surprises. There's no sense raising a child to think that adults never make hard decisions in life, they just follow simple hard and fast rules. Now that we have cell phones, it's a lot harder play one parent off against another in a non-neurotic relationship, anyway.
Also, one of the problems with relying on father's unchallengeable authority and an appearance of authority is that while there are a lot of ways parents can influence the behavior of their children, mystique just isn't the most efficient of these - parenting is too intimate a relationship for an impression of perfection on the part of the parent to survive long. Plus the law now justly limits how much your children can be made to fear your hand, so the more lasting mystique of extreme violence can't be employed as it was only a generation or two ago (very fortunately.)
But don't lose hope, fathers. Nature has given the male an unfair advantage in child rearing. Some tones of the male voice, curiously, are more powerful with children than any formal authority or the agreement of both parents. Namely the deepest tones. Without displaying meanness, obvious threat, sarcasm, anger, bitterness, or extra volume, just a little deep rumble of chest resonance and seriousness in a grown male's voice compels real obedience in children - sometimes to Mom's annoyance: "So how come they didn't listen to me when I said exactly that the first six times!" This seems to be built into humans, and most other related species as well - even birds growl at each other, just at a much higher pitch, scientists have found, and respond to such growls.
Because modern humans live in rectangular rooms which resonate at the lower frequencies human voices can create, males especially learn over time to filter these deep tones out of their own voices and produce a higher, less masculine, and less natural voice. With every generation, this higher tone of voice becomes more pronounced because outdoor work and conversation become rarer. But just put those tones resolutely back into your voice next time you pronounce that it really is bedtime, and you may be surprised at the quick results. This may take some practice, and it doesn't hurt to put a hand on your chest while you practice, to feel for the resonance. Practicing out of doors or in very large rooms may help, since the same ugly resonance won't be there. (This is all true for women as well, but to a much lesser extent, given their higher natural pitch.) Remember, you aren't trying to frighten the bejesus out of the poor wee creatures, just to communicate your seriousness. They'll respond.
It should be noted that parents in neurotic relationships often play out their own dramas through their children, using them as scapegoats or allies. Neither role is appropriate - but such problems between parents should be resolved separately, as they really don't have anything to do with questions of child-rearing. Using children as pawns in an argument they don't have to be part of is exceptionally selfish behavior, and can be quite destructive for the kids. But the problem in such cases is not authority or agreement or mystique - the problem is that two parents are fighting deviously with each other instead of honestly.