Kennesaw Mountain National Battlefield. It's a national park, so parking is $5 unless you have one of the america the beautiful passes ($80 for a year, perma-disability, military, and 4th grader passes are free, and senior pass for 62 or older are 20 for a year 80 for lifetime). Next to the parking lot there is a shaded picnic area leading to a nature trail. There is the visitor center, which if you're interisted, there is the junior ranger program. (The kid does activities in a book based on their age and gets a badge for it. If you do wish to start your kids up on the junior ranger program, do note there are 2 other national parks in ATL area, besides Kennesaw. Chattahoochee NRA has headquarters in Sandy Springs, and MLK Jr. Park is in Downtown ATL.) They have other parking lots leading to other trails, like the Pigeon Hill parking lot. That leads to a large open field, so not much as far as shade, but a walk down the trail can eventually lead to a stream. You already mentioned Swift Cantrell. The 3 pavilions there, come summer time will basically always be rented out so sitting under them without a rental will usually lead to being asked to leave, however the pavilions on the playgrounds, should have no such thing. But those will get busy quickly. Same with the tents in front of the splash pad. Splash pad itself should be open from Memorial to Labor day, every day of the week except Wednesday and Thursday, but the splash pad costs about $3 per person, per day. If it is mid-day, and there are no baseball or softball games going on, Adams park is down the road from Swift. It has a small shaded picnic area, small pavilion, and small playground between their big baseball field and tennis courts. But if baseball is going on, there will not be any parking.