Usually puppies love to play with other dogs but by two they aren't so thrilled about dog parks. I also wouldn't take a new dog to a park. Just too dicey.
But after you get to know your doggie, and would still like to give it a try, find one that isn't heavily used and go at the most off time you can find. Alternatively some parks have sequestered areas for dogs who aren't sure of themselves or are new to the dog park. This lets them be part of the park but not be in the park. Yet another alternative is a doggie day car. Usually the people here will know the dogs and have a decent idea of how to manage them.
Avoid crowded dog parks or those where the owners congregate and ignore their dogs. These kinds of situations lead to issues.
I don't have any experience of dog parks, but from what I read - I would avoid them like the plague, unless they are like the ones frequented by a young friend of mine (with a Basenji and another breed) who lives just outside London. Lovely meadow-like venues, grass, trees and a large area.
Even they can be hazardous - but at least they mirror the countryside and are big enough not to get crowded.
Like others, I am not a huge fan of dog parks and have had an unpleasant experience with an unpleasant GSD owner who felt he should kick all three of my dogs who were trying to get his on-leash GSD to play in the off-leash park. I still value dog socialization, though. There are other ways. You might visit a local dog club to meet owners who might agree to go on walks with you and your b in neutral territory. You can walk in the area of a dog park and again, perhaps meet an owner who is willing to go for walks. After making friends with a buddy outside of the park, meeting at the park when there are few other dogs around for off leash play may make the park intro pretty simple and less intimidating.
I took my male to one when he was about 2. First went in the small dog area (where there was only one other dog) but the grass was so high I was afraid of ticks. Then went to big dog area and mine just wanted to sniff the fence area until some bullies came along and started to really pester him - he took off for the far corner (thank goodness). Forget the dog park.
A dog park sounds great, but in reality...
I think this largely depends on (both) the owners and dogs that are there. Many people are responsible and just want their dogs to have fun. Plenty of people aren't. Get to know who goes there and you can avoid many of the problems that give "dog parks" a bad name.
The first time I took doodle, she started shaking -- too many dogs and she was the "new" one. So we left. I took her back when it was empty and allowed her to roam the area, smell, and get a feel for it. Then she understood. Lots of different dog smells. This wasn't anyone's yard or territory -- it clicked. The next time we went there were just a couple of dogs and her reaction was totally different. She loved it! And still does!
Are there "bad dogs" there? Some, but in general this has been a very positive socializing treat for doodle. We don't go every day, but 3-4 trips a month seems like a good balance (to keep it fun and not let her think she's in charge there).
I always try to make sure doodle gets to do something special. A (really long) walk, off-leash time in the woods, the dog park, or down to the water (we live near the ocean)... something random so she never really knows what's going to happen, even if it's a day when we just chill.