Derek,
It is quite simple. At the park, the dog can run up to any other dog and mark where ever it wants, thus setting up its terratory. On leash, you are in control, and the dog is resisting your leadership, displaying dominance and aggression towards other dogs from a distance (hair up, slowing down walking speed or even freezing in place, slow tail wagging, ears back, as the other dogs approach closer, it escalates into pulling, barking, growling...). Frankly, by taking the dog to dog parks, you are worsening the problem. With a prong collar, you will want to correct the dog HARD at the first sign of such aggression (which is why you will want to walk the dogs separately until you get this under control). Here is what I suggest.
Find a dog in the neighborhood that is tied up or behind a fence that will bark and act wildly as you walk by. If you have an e collar, GREAT, if not, you can try it with a prong, but for some dogs the prong will agitate them more if you can't really effectively correct them. Then start from some distance away with your dog in a heel. The second your dog looks at the other dog, say NO (or HEY, which ever you use) then immediately give a strong pop with the prong collar, keep walking. At the same distance, test your dog again. If the dog again looks at the barking dog, you can do one of two things, either back up because you started too close, or pop the prong again, but harder. Test the dog again, and don't move in closer until you get an avoidance response from your dog (she should look away, or look at you or look straight ahead). Then move in closer, and repeat this exercise until you can walk your dog right next to the other dog and get no response from her whatsoever.
Leash aggression (aggression shown on a leash) is fairly common and the root cause is lack of control on the owner's part. By doing the above exercise, as well as other dominance building exercises, you can get control of your dog when other dogs are around. To be honest, although you claim that both dogs show limited aggression in the dog park setting, the aggression is there and could be an accident waiting to happen. When you say your hound will bark/growl if another dog gets on her nerves, that means when another dog trys to dominate her. If the wrong dog (another dominant aggressive dog) approaches a dog like your hound, you may be in for a proper dog fight, in fact I would bet on it. Your dogs don't need dog parks, just walk them regularly, train them regularly using obedience and the basic principles of pack dynamics and their behavior will improve, you will see. At the dog park, you completely lose control (unless you have e collar trained your dogs, which I assume you haven't). Dog parks are not safe places for your pets and lead to all kinds of behavioral issues like the ones you are seeing with your dogs. I see it ALL THE TIME in my clients dogs.
In my opinion, the addition of the puppy renewed your hounds interest in leading your pack, and going to the dog park enforces her role as pack leader because you can't enforce your commands AND there are so many other dogs there for her to dominate and build her pack drive. The aggressive behavior on leash is a very typical behavior of a dog that does not respect the human as the pack leader. This is easily, easily fixed if your dog is not handler aggressive and you can invest the time in rebuilding your pack, with your dogs at the bottom!!!
Good luck.
Michelle