Frick Park in Squirrel Hill has a lot to offer for calisthenics. Even a lot of hills to walk or run up. That's not exactly on the North side though but it's so nice it might be worth the extra travel.

They've got pull up bars with varying grips and a platform that even lets you do inverted rows, parallel bars, chest press, ab things, etc. Worth going to at least once to check it out. When I first moved here it was my haven. Plug in 'blue slide park' in your GPS and just follow the paved path into the park and you'll find all of the equipment. Also do explore the dirt paths when it's a little less muddy, they are lovely and would be a great workout if you like to run or even walk.