So let us both be clear on what we mean

Unless you want to point out where someone has actually stated they support the State in exercising authority in favor of parents, you are simply putting words into people's mouths. You are attributing to us an argument that was never made.

If people argue in favor of parental power and authority, then clearly it can be safe to assume that they expect some sort of external power--most likely a state--to back up this power for them, otherwise they would be unable to force kids to submit to their every whim. I think it was important for people to explain that they did not expect the state to enforce parental authority, because clearly that is how parental power and authority is all but insured today.

It is completely wrong to claim parental authority could not exist without the state. Of course it can, a child can submit to the authority of a parent without the State interfering. Why do you find that inconceivable?

I do not find such a thing inconceivable at all. After all, there are some women who willingly submit to male authority in the BDSM community. If some kids chose to willingly submit to parental authority in a youth liberated society, I would have no problem with it, because it would be their choice. I am simply saying that without a state there would be no way for parents to force kids to submit to their will or to remain dependent on them until a certain arbitrary age if the kids in question desired their emancipation, and I figured that it was clear what I meant. But it's instead clear that both sides of this issue needed to explain exactly what we meant to the other side. It would seem we both made assumptions that we shouldn't have, but neither of us were fully clear.