In every state, the owner or possessor of a dog having a dangerous propensity (i.e., the propensity to jump on people or bite people) is strictly liable for injuries that result from the dog’s dangerous propensity. So, if a dog owner knows that his dog likes to jump on people and knock them down, and if that happens in a leash-optional area, the dog owner will be liable under state law.
Also, most states have laws imposing strict liability on dog owners whose dogs actually bite someone, irrespective of whether the dog previously bit anyone. These laws still apply, even though the bite occurs in a dog recreation area where leashes are optional.
If a dog owner knows that his dog has the dangerous tendency to attack and fight with other dogs, he should not bring that dog into a dog park. Doing so would be not only negligent but would constitute a reckless disregard for the safety of other dogs, the rights of other dog owners, and the safety of other dog owners. A dog owner should be held fully responsible for all resulting injuries to other people and their dogs. The doctrine of assumption of the risk should not apply because the victim cannot be regarded as accepting the risk of such negligent, reckless and possibly illegal conduct.
To find out how to investigate whether a dog has any kind of a dangerous propensity, see Investigating the Attacking Dog.