Auto-syncing of passwords is a core feature of every modern browser. Safari does it, Firefox does it, Vivaldi does it, Edge does it, Opera does it, and, of course, Chrome does it.
Want control over your child's account? Google has something for that: https://families.google.com/familylink/
Want the school to manage it for you? That's exactly what G Suite is.
Edit: The problem here is ignorance. It's easy to demonstrate that it's possible to control all the data about which this parent is concerned. Sure, there's data they can't easily control, but they make no mention of that. Your interests, demographics, everything else Google infers about you... you can't fully control that, and that's what parents should be worried about--not password syncing.
Many of these claims are outright false because they’re technologically infeasible and would require the exploitation of security vulnerabilities. Companies like Google are constantly having their apps inspected by hobbyists and professionals alike; there’s no way that’d go unnoticed for any length of time, plus it’s nearly impossible to begin with.
In short: a parent has just discovered the internet. They’re scared, paranoid, and confused. This is, unfortunately, quite normal.
There are legitimate privacy concerns regarding Google’s collection of data, but none of them are raised in this article. It’s a misunderstanding at best and outright propaganda at worst.