“colleges can tell from a student’s profile what kind of educational background the parents of a student have”
No, you can’t tell about the parents from the kid. But what was said early on, is you can tell how far a kid has come from where he started.
Top schools aren’t admitting the parents. They’re looking for energy in the applicant, his or her vision, dedication, and industriousness, etc. Just being born into some money doesn’t mean a kid has it. And there are many programs and influences on lower SES kids that do foster what some parents can’t.
Being wealthy provides opportunities and a safety net. It doesn’t guarantee a kid stretches or climbs out of his box.