You’ll definitely need a plugin for that. It’ll need a custom field for the parent address. Also, do you think you want to send the email when they log in, or when they visit the site? Users typically need to log in only once unless they make a point to log out. If someone told you that the email needs to be sent when kids log in, you should not tell them what I just told you.
Since you are the first person to ask for this, it sounds like bad legal advice to me. But there is a long history of schools following bogus advice.
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.
This is not exactly what you are describing, but you may be able to make it work for your general requirement (the concept of following and notifications, between relationships):