Justin and I (your hosts here at
WildTexas.com) live in San Antonio. You can filter the search results here on Wild Texas to show only parks in the vicinity of San Antonio --
San Antonio, Texas Camping, Parks & Lakes
We concur on the recommendations to Government Canyon SNA, Lost Maples SNA, Pedernales Falls, Hill Country SNA and Enchanted Rock SNA. Guadalupe River SP also has a nice long (VERY ROCKY; can be a little annoying by foot or bicycle) mixed used trail. The trailhead for it is on the right literally just after you pass the ranger station.
There are also small local city and county parks that may offer opportunities (paved and unpaved), including OP Schnabel Park, Friedrich Wilderness Park, the new Leon Creek Greenbelt by UTSA, Walker Ranch Park has some natural trails as well, though mostly paved.
Apologies for not having more of our local parks featured on WildTexas.com. I have had very little time to devote to adding content and photos to the website since my little one was born 17 mos. ago.