The parent may genuinely be writing at the students request.
It’s seems reasonable there may be at least some students who genuinely want to do more work to improve their chances of being accepted in to their preferred choice at the next level.
I am assuming kid will struggle in the top school anyways because of parents not being able to control as much anymore and since self-motivation was never trained it will be hard to do things on your own.
You might say that parents will use same approach in top school as well, but if they had that kind of power I guess they would not need to put their kid into a public school in the first place.
I guess it's not impossible that a parent would intercede on behalf of their child, with the child's knowledge and permission, it just seems unlikely to me. Any student who is actually contemplating Princeton is almost by definition extremely self-motivated and independent.
This is how you get kids with parents who go to job interviews with them. I actually had to teach a kid I went to undergrad with how to grocery shop because "I never had to do it myself"
Kids need to learn how to be their own advocates.