No, you misunderstand what the logic here is. Parents should be able to consent to some things in the name of their children and absolutely should not be able to do it for other things (even those things their children would be able to consent to were they grown ups, so I don't mean illegal things).
Exactly what stuff the parents can consent to is the real question and should be argued properly and not treated as any kind of "common sense". Things that are necessary for the safety of the children - like vaccines - yes. Other things are on much more shaky grounds.
I perfectly understand the logic. Data processing is ubiquitous these days and control and consent falls well within what parents are there for. That is common sense: parents are there to make decisions on behalf of their children on issues children cannot handle.
This is not a perfect answer either. There are many situations where a parent is at odds with their children as to what is appropriate. In some cases, the parents should "win", but in some cases, the child should "win".
Parents should be able to consent on behalf of their children and retraction should come from parents as well.
This used to be considered common sense but not anymore.