The short answer: No.
The longer answer: It's a dog park, which means it's primarily for use for people with dogs. If it's an off-leash dog park, it's intended for dogs to be able to run and play, and a running child could easily be knocked down, possibly injured.
In this case, the parents are being very entitled and special snowflake...surely there are playgrounds or areas where dogs aren't allowed that could be used by the children instead? Besides, why would anyone let a toddler, who is as likely as not to put dirt or anything else in his/her mouth, run around in an area dogs have been using as a toilet? In an area where dogs have been doing their business, it's quite possible the kid could pick up a parasite or something. Seriously, what are these people thinking?