How does giving Apple my CCV code confirm I am a parent of my child?
It confirms you are owner of the card - or at least have it in your hand.
Presumably, that is considered a minimum identity check, assuming the last names match. Sending in a photocopy birth certificate seems a bit 1980s these days ;)
The CARD Act of 2009 says that to have a credit card in one's name they must be 18 years old with a verifiable source of income or be 21 years of age.  Asking for the CCV of the card when setting up the child's account is just Apple's due diligence that the card associated with the parent's account is &quot;in hand of an adult&quot; and that the parent is aware the account is being created.