I'll chime in with a couple of thoughts/comments on the parental consent idea.

First off, I'll mention that cases of abuse (actual) came to my mind when reading your post, cases where parents have involved their children in sexual activity. I know that a difference is that your model is based on the child consenting to the sexual activity, but it can be a reminder that parents having a say is not a guarantee for things being okay and there being no harm.

As I understand it, your model is an idea for a law. As opposed to open-minded parents with sound judgment allowing consensual sexual activity discretely, thus risking problems with the law if it is found out.
  If the law were put into effect now, the difference would be that there would a law to support parents, instead of child protective services taking their kids from them (not to mention the trouble for the adult and the kid), if found out by authorities. Compared to today, though, in most countries, it would be less forbidding by the state/government.

As compared to parents deciding if the child may drink alcohol or date a certain person, it could be a good thing, if the parents can see that the sexual activity is a bad idea or the adult being a bad person. One could then also discuss strictness and rebelliousness.

I have come up with a couple of complications, which I'll mention.

What if the child becomes involved in sexual activity with an adult without the parents knowing about it, and the parents have a sex-negative reaction without respect for the consent of the activity. If only the law were changed, how many parents would then report the adult? And if the adult were reported, would the court then respect the opinion of the child or judge by the absence of parental permission?

What if the parents' judgment is clouded by the status of the adult involved? What if the adult is the dad's boss, and he can't risk losing his job, would it then take more for the parents to disallow the relationship? For the sake of argument, we could say that the child doesn't feel strongly for or against the sexual activity.

Also I'm not so sure that regret and shame will vanish. Perhaps there will be cases of adult children having regret, and then blaming the parents.I think parents having openness about such subjects as sexual acitivity, and acceptance where it is due, is important in mitigating regret and emotional issues. That can exist with or without a 'parental permission law', and a 'parental permission law' can exist with or without parents having the mentioned behaviour. The law might, however, move public attitudes in a healthy way.