I feel that anyone bringing a small child to a dog park is taking a HUGE gamble with their child's safety. You know your dog. Do you know everyone else's? Are you willing to bet that in a large dog park, every single dog there is tolerant of kids? Especially if your child is not yet old enough to understand that not all dogs are like their own. Tragedy can happen in the blink of an eye, and it's often the dog that pays with its life if it bites a child, regardless of if the bite was unknowingly provoked or not.
I've witnessed several incidents at the dog park involving young kids. One small boy about two years old was toddling along next to his mom and their golden retriever. An overenthusiastic young boxer knocked the kid over simply running by and bumping him with her back end. The child fell down, whacked his head on the packed-dirt path and started to wail, which triggered the golden retriever's protective instinct, and the golden and the boxer got into a fight. The child and both dogs were injured and no one wanted to claim responsibility. The child getting knocked down was a complete accident; the boxer hadn't even jumped on him, just bumped him on the path and toddlers are unsteady. The golden retriever was doing what he thought was correct by protecting his family. The boxer was defending itself from the golden retriever. It was a mess.
Another incident involved a child who tried to excitedly grab my friend's Papillon from behind. It was a cooler evening so Winnie (the dog) was wearing a little pink parka (I swear my friend had more clothes for the dog than for herself) and so I can see that in the eyes of a small kid, Winnie must have looked like a living stuffed toy. The girl squealed and tried to grab Winnie from behind, who of course was startled and tried to nip. The girl was unhurt but Winnie was frightened and the mother was furious. Especially because SHE (the mother, not my friend) had told her daughter it was perfectly okay to go up and pet Winnie, without asking the dog's OWNER first. Though IMO the blame for that incident falls squarely on the mother.
A few days after the above incident, we were back at the dog park again and because it was later in the season (October-ish) and got darker earlier, my dogs had their light-ups on their collars (Just LED globes that attach to the dogs' collars that either light up solid or blink, to make the dogs more visible in the dark, for both other people, other dogs, and me). We were trying to leave the park and actually I had just finished observing to my friend how popular the collar lights are...in the dusk the dog park looked like a mobile Christmas light display...when I heard an excited shriek, followed by Radar and Jagger (my two dogs) bolting in opposite directions. A small child was overexcited at the sight of all the moving lights and was screeching and running up to everyone's dogs and trying to grab the lights. Radar was startled and upset by the noise and had taken off out of fear. Jagger, on the other hand, was just as excited as the kid and in his mind the game was on. I managed to track Radar down and it took me considerably longer to convince Jagger that the game was over and it was time to go. Every time we made progress toward the parking lot, the kid would come back, shrieks and all, and the chase would start all over again. After two or three attempts to catch Jagger, I put the leash back on him (Radar already had his back on) and I must have been visibly annoyed, because the kid's mom called after me, "He's just a KID and he's just PLAYING!!" By that time I just wanted to get home and have dinner, so I didn't say anything, but in hindsight I really wish I had done. I don't know what happened after we left the park, but that mother (and son!) will have been very lucky if that child was not bitten. Running around and shrieking and acting like prey in a large field full of predators is a very unwise thing to allow your child to do.
The dog park is a public place and everyone has a right to be there. Parents have a right to bring their kids along as long as the rules permit, however I'm going to have to disagree with the idea of bringing toddlers to the dog park. It's just an accident waiting to happen.