They wouldn't do it for the money — they would do it to force the kid to take down the photos.

I think the GP's point is that there's an implicit "or... what?" here. The parent would sue the child to either take down their photos, or... what? What would be the consequence if the child didn't comply with a ruling against them? The only potential legal consequences would be on the parent themselves.

In effect, the law has no leverage against children, except insofar as children can be influenced through their parents. If the parent were the one suing—because the parent also has no leverage—then nobody has leverage.

Parents usually have plenty of extra-legal leverage. That's why such a "case" wouldn't go to court, just as a dispute over the chaos in the child's room wouldn't. Application of that leverage is called "parenting". :-)

Yeah, that was my point. If a parent feels that they have to sue, i.e. if they feel the state's assistance is necessary to get their child to comply with their wishes—then that's because they're in a situation where parenting isn't working / they don't have any of their own leverage over their child "left."

And if they're in such a situation, then what do they expect the state to do? The state, in the end, only relies on their leverage. (Except in certain exceptional circumstances, like if a child is reported as being suicidal.)

I don't know elsewhere, but in Spain social services can take children from their parents if they're not being taking care of properly. And that doesn't necessarily mean that the parents are to blame. So if the parents can't control the children, they can tell government: we can't hold them anymore, their behaviour is antisocial and we can't take responsability if they commit a crime.

If the parents are unable to coerce the children without help from the legal system, they're expected to call social services to take over.

This is not a hypothetical case, older minors can overwhelm physically their parents (or grandparents, usual when parents are drug addicts) and sometimes the children are very difficult to deal, maybe they have a personality disorder.