> Is it a reasonable concern?

Yes.

> Parents are supposed to raise their children.

Yes.

Vapid questions and statements aside, ...

Here is just one solution that helps parents, and respects everyone's privacy: Zero knowledge proofs.

Which allow anyone who is verified by anyone already (i.e. credit card company, ...) to get a cryptographic key from that organization, that they can use to anonymously verifiably assert they are 18+ to sites, (1) without giving sites any other information, and (2) without their key source getting any information on what sites they visit.