As an adult that has grown up with these infections, I think you gave the right guidance for treatment. These infections have robbed me of many, many things - a price that no child should have to pay.
Have them read some of these threads, or share your story with them again, or share this link to help them understand what they are preventing their child from going through with this short treatment:
www.tiredoflyme.com/so-you-have-a-friend-or-family-member-with-chronic-lyme-disease.html#.U3dTc3bbS1t