It's arguable. The police likely wanted to give a citation for a noise ordinance violation and without your ID he could not. Courts have found that the right against self incrimination does not extend to biographical information on an ID. So it depends what the arrest was for (resisting?) and specifically what happened, but there are times where a police officer may arrest for a refusal to show ID if they have a lawful purpose for requiring it. And yes, police can and do interview children without parental consent, although that fact may be considered in whether the child freely and voluntarily spoke to the officer. Hire a defense attorney to review all the facts and allegations.
This is not to be considered legal advice nor does an attorney-client relationship exist.