So many children's dog phobias are caused entirely by out of control "friendly" dogs bounding around with blithe feckless owners who accept no accountability for their pet's actions.
I'm confident with dogs and grew up with giant breeds. My DCs unfortunately don't have any friends or family with dogs that they can build a relationship with.
It's getting better now my DCs are getting taller than the average dog. Around a calm dog, DS2 can now ask if it's OK to pet it. DS1 tends to quietly shy away. But an energetic dog instantly alarms them because so many times as toddlers they had dogs bounding up to them, licking their faces, attempting to steal food and bowling them over while the owner tries to tell us that it's OK because their uncontrolled fluffball is "friendly" and loves children and completely fales to take a subtle hint that howling or fleeing children don't love dogs 
I challenged a dog owner to put her dog on a lead after it jumped up at me for the THIRD time. Apparently I'd been scaring it when I firmly told it to get down 
I don't have dogs, I'm calm with them, don't eat while I walk etc, there is no reason in my behaviour to encourage a dog to jump up at me, yet it happens surprisingly often.
If a dog is out of control, the owner is morally and legally in the wrong.
I like dogs. It's just a shame about a not insignificant minority of irresponsible owners that ruin it for everyone else including sensible owners.
OP's dog owner was in the wrong for the dog's presence in the play area, the dog's behaviour and their shoddy attitude to the situation.