Beagle barking while playing with other dogs

I'm sure others will have better, more comprehensive solutions, but I would say that since the reason he is barking is, as you said, being over-excited, the best thing to do is work on calmness around other dogs (of course). You could start by taking walks with one or two other people with their dogs to get him used to being around other dogs, once he seems totally calm with that, you could try simple training near, but outside the dog park. Get as close as you can to the dog park without him getting crazy excited, and work on basic commands like sit and watch me. Slowly work on moving closer and closer until you are right outside the dog park, then eventually, if possible, inside the park. Also, try having him meet the dogs through the fence before going into the dog park. If he begins to bark, move him away, and have him calm down. Then try again. It may take a while, but eventually he should start to calm down a bit. Hope this helps, and good luck!

I hate dog parks. They are dangerous. They can be a hotbed of disease.

It is far better to know someone else (or two people) with a dog and see if your dog and that dog can play together and set up play dates.

Regular play with the same dogs is far better than the unpredictability of a dog park.