Hi there! I'm sure your parents are very excited for you to get a job!
Here's the thing though, they are not going to go to that interview with you, they are not coming to work with you, they aren't the one's that did the work that made you someone to consider for the job.
I am honestly not sure why a recruiter would ever come into contact with your parents, but if that's even a possibility in your current situation, I would strongly recommend letting your parents know it is time for them to take a step back. If we are talking helping you make connections, network, get your foot in the door, introduce you to people who can help you...parental help is great.
But the tangible actions (job searching, writing a resume, applying, communicating, interviewing, following up) that should be all you, not your parents.
So in answer to your first question...parents might help you search, but they should not be involved past that. No recruiter is hiring you and your parents.
If you are under 18 and require permission from a parent or legal guardian to work, you should consider seeking information about legal requirements from the Employment and Labor board of the state where you reside.