That might fit if I had said that parents' wishes ought to be obeyed because they have the right to do what they want with their kids. But any role I have suggested for them is indirect, based on the idea that they are in general good judges of what's in their children's interest.

If we are concerned with a specific kid, then I think parental perceptions are very likely to be important. If my 13-year-old daughter was totally infatuated with a 14-year-old bad boy who shows no real interest in her, and he suggests they meet for sex, I'd strongly urge my daughter not to. If the same daughter has sex anyway with that guy, and another similar one and recovers from the emotional aftermath, and then meets a sweet 14-year-old boy who seems to really like her and she wants to have sex with him, then I might say it's fine with me. Maybe even that it's a good idea (assuming it's legal, which I think/hope it still is in most places). It all depends on context.

If we are talking about societal policy, then I think several things are relevant, though none decisive. One is what 20-somethings wish they had done or not done when they were young teens. Another is what parents as a class think -- if the vast majority think sex with adults would be bad for their kids, then that's some evidence in support for not making it legal and accepted.

I have never seen any evidence for the claim that parents want to control their kids' sexuality for the sake of controlling them more generally or getting some personal benefit. As far as I can see, it's all for the long-term benefit of their kids as they see it.