My puppy is really submissive/afraid with other dogs. Low back, tail between the legs, turn on his back when other dogs come to play with him and won't let any dogs smell his butt. That being said he absolutely loves the dog park. He runs toward it, become crazy excited in the car when we go near it, wag his tail like crazy when inside the park. He goes to every dog to sniff them. He let dogs sniff his face and his privates, when a dog owner pull his dog away from mine, he will run toward the dog to continue playing.
I can tell he wants to play, he will run toward other dog and then turn around in a "chase me" style, but if it works he runs toward a safe spot (a human, a fence, under a bench...) and play with the other dog there.
My dog is small (Cavalier King Charles Spaniels, ~15lbs / 7kg) and he is 6 months old. His behaviour is the same no matter the size of the other dogs, no matter if they are exited or really calm.
What should I do to increase his confidence. Any thing I should do/avoid? When he is playing I never pick him up I watch closely and in the rare occasions where the other dog is "too much" I will gently push the other dog away, but that doesn't happen often.