This park came up as being a calisthenics park. Do they have a trail that includes parallel bars and pull up bars and such?

There's a fitness trail with exercise stops along the way. They're beginning to show their age, but they're there and useable

There is a path around the lake. All along the trail there are various activities, climbing, stretching etc.

Yes

Yep, nothing fancy but it does the job.

Yes and I heard it is being upgraded.