My mom would drop me off at the office with "her half" of the amount due once I was old enough to make sense of conversations with doc but couldn't drive (I think around 13). That way, they got paid her half and would bill my father the other half, which they did, kindly. Before that, my mom always brought ONLY enough cash to pay her half in with her.
When we went to our family doc who we'd seen for YEARS, that wasn't necessary, but for a while our insurance changed and we had to see a "stranger" who was not happy about billing my father for the balance. They wanted our business, though, so because she brought half (I think plus a dollar) in, they saw me and billed dad for the rest. If he was good about reimbursing, my mom, the custodial parent, would have been happy to just have the amount billed and collect from him. BUT he wasn't. So, he was listed on the paperwork as the "responsible party" and my mom brought in half of the payment.
THen, doctors started to change rules and the "responsible party" had to be the one to bring in the ill child. SO my mom DID actually just start calling him to pick me up and bring me to the doctor...again, because he was terrible at paying medical bills and we didn't have enough money to eat, let alone pay doctor bills that were his responsibility. Of course when he picked me up, my mom would have her half in cash in my pocket.
Silly, the whole thing. I wish everybody could have just been adult about it.
So...I think it worked that way, though. I think you should call the doc beforehand for an estimate of the cost for the visit (assuming a culture or blood test or whatever they see as protocol for the presenting symptoms), and then let the other parent know that you will bring your half with, and will expect that they will do the same.