Most of the city parks in and around S.A. are not open after dusk (such as
Friedrich Wilderness Park and
Eisenhower City Park). Those that are open after dark are probably too well-lit to afford skywatching opportunities.
I live on the northeast side, less than 1 mile outside of Loop 1604 on Blanco Road. We have a surprisingly good view of the sky from our suburban backyard, thanks in part to Camp Bullis (U.S. Army training area) being located just across the street from our tract. We still have all the light pollution from S.A., of course, but the bulk of our view faces away from the city lights. Meteor watching is spotty, so I wouldn't say it's really DARK out but no trouble seeing enough stars to remember what they look like... which isn't bad, living in a city the size of S.A. There's a county park just down the road from us (Bullis County Park -- I don't have a park profile posted for it, yet), but it's closed after dark.
Back when Halley's Comet was last around ('86, I think it was?) my parents and I drove out just a ways into Helotes... well, Helotes has grown considerably in size and is practically (if not in actuality -- which it may be, for all I know) a suburb of San Antonio now.
You might try one of the
Canyon Lake area parks (Comal Park, etc. -- note, some have entrance fees that are comparable in cost to Texas State Parks) or, if you don't mind the drive, head out to Bandera, Texas and
Hill Country State Natural Area. The campsites there don't afford great skywatching (dense tree canopy), but just a couple hundred feet from your tent's "porch", you'll remember what a dark night sky looks like again.
Guadalupe River State Park is close to S.A., but I've found it doesn't offer much sky viewing opportunities due to dense tree cover and the steep, rocky bluffs down by the river. I s'pose an after dark hike (with red-filtered flashlight) on their relatively new multi-use trail would provide an opportunity to find a good vantage point, but can't vouche for that as I've never tried it.
And
Big Bend... well, let's just say one of the primary reasons I didn't want to leave Big Bend NP when we visited was because of the night sky views there. You can swim in the stars out there!