I wouldn’t be going back to that dog park ever. OP said she would pay for a puncture wound. If the dog was treated for a puncture wound, then she should probably pay. If it was something else, I wouldn’t. I would want to see the vet report before paying anything.
There is a dog park near a walking trail near me. It has two enclosures, neither overly large, maybe 150’ long. One side for large dogs and one for small. We have let my girl play there on occasion. We watch the dogs that are in there already playing, looking for red flags before we go in. We haven’t seen any truly aggressive dogs, but several times we have not gone in due to the play style of a particular dog or two.
Only once have we left early, when a woman brought a dog in that really shouldn’t have been there. We called her the “fun police”. The dog wanted to control all the other dogs and tried to scold other dogs that were playing together nicely. The lady had her there for “socialization” although the dog clearly did not want to participate in play with anyone.
Thankfully my current girl has excellent social skills with other dogs and doesn’t react or escalate when another dog exhibits bad behavior. She has dog friends that come to the farm to play, but at times, we go through a dry spell when no one has time for a visit. Those times it is nice to have the option of the dog park, although we have only gone a handful of times. Oh, and she picked up fleas there the last time she went.