This happened to me twice. Once, I was in Costco and when I left, a lady came running over to me and told me she was Costco security. She said a guy was with his little son, the son was pushing the shopping cart and lost control, and the cart hit my car. Long story short, she got the father’s info, I got in touch with him and he wanted me to take the car to one of his friends’ shops for body work. Luckily I was with my mom who advocated for me at that time. She said no, she’s (me) not going to take it to your friend, you’re going to pay her what the estimate says (which was $400 – this was about 15 years ago). The dad gave me the money. I forgot at this point whether it was cash or check. Probably cash.
About two months ago, I was in a fender bender where I hit the car in front of me. The details are too tedious to tell here, but it was definitely my fault and I was very apologetic. A few phone calls back and forth with the lady I hit and she got an estimate from her body shop (about $400 or something similar), I met her at a Dunkin Donuts, gave her a check and was very grateful she didn’t decide to take me to the cleaners with her body work. She had an Escalade that was a few years old. The damage was relatively minor but if she were unscrupulous, I would have had a fight on my hands.
I’d say for you, @hominid, talk to the dad and see how he wants to handle it. He may say get one estimate, he may say get two. Whatever. He shouldn’t give you a hard time about the money. It sounds like he doesn’t want it on the son’s record, or maybe, since the son is a new driver, he doesn’t want his rates to go up, or maybe his insurance really isn’t for a driver under 21.