In encroaches on parent's real/perceived authority

In the case of sex which causes pain, the answer is obvious.

In the case of gentle, caressing, loving, enjoyable sex the answer is more subtle:

They don't call having sex making love for nothing. Having sex can and in many cases does forge a tight emotional bond between the participants.

Parents don't want their precious little ones from loving anyone more than they love them. They view their children as their chance to pass on their ideals, morals, values, etc. to the next generation.

Unless their values tolerate or encourage intergenerational sex, such as some religious groups like "The Family" and some Australian Aboriginal cultures did or do, exposing their children to the very idea that "sex between children and adults is okay" is offensive to the parents: The parents will view these ideas as factually incorrect and will have to spend time reeducating their children to believe the Correct Way.

Of course, actually showing the children that sex between children and adults can be not only okay but enjoyable makes it that much harder for the parents to convince their children to see things their way.

The reality of politics means that in most cultures, parents have great autonomy to teach their children whatever they want, even if it's factually wrong. This means the state generally frowns on anyone forcing someone else's children to "learn" something the parent's don't want them to learn, unless of course it's the state or a powerful lobby group like Madison Avenue who is doing the teaching.

There is an element of The Golden Rule here:
How would you like it if you were vegetarians and you taught your children that eating meat was evil, only to find your neighbor who knew your beliefs not only teaching your children that eating meat was not evil but giving them hamburgers behind your back. The average parent in this situation would be furious. Substitute typical sex-fearing parent in this story and you'll see the real reason parents don't like the idea of child love: It's an affront to their role as parents.