I don't think they have any formal authority, but there are often reports of them trying, most often trying to enforce helmet-wearing. It's probably an area that hasn't been explored properly, and it would be interesting if this were tested in a court of law.
Do schools have any authority, legal, moral, or otherwise, to say how a child travels to school?
Do Local Authorities?
Sounds like overreach to me.
I do hope the parents just tell them to fuck off.