We also have some nice small parts include Friedrich Wilderness Park (has a couple wheelchair accessible trails; no bicycles permitted) and O.P. Schnabel Park (awesome, including some technical, mountain biking trails + playground, etc.); neither offers overnight use, however. I really need some time to update WildTexas.com because San Antonio's local park inventory (city and county parks) has dramatically increased in the past couple years. It's almost as if the City has actually realized for a population its size, it needs more parklands and public spaces *gasp*! Maybe we are learning a little from Austin, after all. ;-)
Guadalupe River State Park has some great hiking, mountain biking and even equestrian use trails that are often forgotten because the trailhead is literally RIGHT after you pass the park entrance station/HQ. Be prepared for a rough ride, though--most of the trail is very rocky. There are
RV and primitive sites in the park and lots of them, so enjoy. It's really the park we most often end up at since it's most convenient from where we live and we often are so long out of camping we need a close-to-home park to check that all our gear and planning is still camp-ready. That and having a toddler (last time we were there Justin had to take our kiddo home at 2 AM because she was having a complete and total meltdown. She's older now and should fare better.)
Hill Country State Natural Area in Bandera gets forgotten a lot too but if you love solitude, lots of trail mileage and don't mind a few horses and/or mountain bikes during your hike, you'll love the park. Some primitive hike-in sites (no water) and some slightly more developed (still no water) sites. My favorite park for feeling like I'm truly getting away, but haven't had a chance to get back there since our youngling was born.