Primary (parents) can set policy on secondary (child) mailboxes such that secondary only receives communications (email, IM, invites, etc.) from entries in the child's address book (whitelist).
Communications from anyone not in the child's address book would (optionally) be surfaced in the primary "inbox" for the parent to approve (and add to the child's address book) or deny (no such add, communication discarded).
In this mode, child cannot add user's to their own address book without parental approval.