Better to ask for forgiveness than permission with RC cars in parks, nobody will want to be liable if you hurt someone so their safest response is to say no.
As to whether you get fined, even if there are park rules against what you're doing, just use common sense. Generally most skate parks will prohibit RC cars, if the park is empty then chances are you won't be harassed, but if it's crowded and you're driving dangerously close and/or being a nuisance, then expect to be fined and kicked off the property.
My dad used to be a park ranger in SoCal and skateboards were the biggest infraction that he had to enforce. Most kids simply carried their boards when asked and nothing came of it, but there were some kids who ignored him and were combative, those were the one's who were fined and in some cases were arrested!
Be courteous and use common sense and things will go fine, do stupid things... win stupid prizes!
If your car is tearing up the turf and causing any damage, then expect to be responsible to pay for repairs if you get caught!