Maybe you could create a custom user field for “parent email address” that people can fill in as they set up their accounts. You can then follow up to get in touch with their parents before approving the account? If your community is for a small number of children and their parents, and you pretty much know everyone already, this should be manageable.
For a larger community where you can’t verify these relationships directly it can be tricky to come up with a workable process and then actually implement it. But yet, as Jay says.. you’ll need a plugin. Or SSO with another platform that handles login.