How is it the parents doing this, I didn't see it written anywhere in the article that they're lawyers.

The 12 year old is the plaintiff, not the parents. The lawyer may have thought she had a good enough case to hold billing, and may have even held disbursements (although more likely the parents agreed to pay for disbursements for the case).

Do you have a message about that you're not articulating, or is this just some good 'ol pedantry?

That would be the ACLU taking the case on her behalf, not her parents doing the suing.