Oh, boy, I have some thoughts on this, thanks for asking! I wish I hadn't taken my puppy to the dog park when she was so young. One of the parks here has several individual runs, and one community park. I took her to the individual runs often, but one evening they were all full, and there were only a couple cars at the group one, so I thought we'd try it. She was maybe 16 weeks, after all shots anyway.
Two huskies, with no owner in sight, came galloping up to us and just kind of swarmed around her - in no way aggressive, but for a little puppy, totally scary. I was too stupid to pick her up and show her I had her back, I just stood there like a fool saying "good girl Jedda" in a happy happy voice, like that was any help. I still feel so guilty thinking about it! She had her tail totally tucked up and ears back and was just trying to get away from them - they were simply checking her out, but she was really scared. If those two were anything other than curious, I could have ended up with a dead puppy.
This by far wasn't the only mistake I made, but for a while she had some fear aggression issues I had to work on a LOT. She still is scared when two or more dogs approach. We sometimes go to a really large dog park (because it's such a great place to hike off leash) where I can easily see what's approaching, and I just give other dogs a wide berth, especially two or more dogs. I avoid like the plague the flats where owners congregate and let their dogs run amok.
I wish I had read this before I got my pup: Leerburg | Socializing Puppies
When your post says that you specifically want a dog you can take to the dog park, my advice is to wait until your dog is an adult, and pick the park REALLY carefully. Good luck and have fun!