Second week in April through second week in May, +/- one week. I would suggest Inks Lake SP (always do) but Guadalupe SP is closer to San Antonio and also very nice. Both are very close to Hwy 281 and gives you great access to all areas of the Hill Country. No matter where you want to end up, get your reservations soon! Hill Country parks fill up fast in the spring.
Short on time at this moment. Will look at this thread later and will be pleased to answer any other questions I can.
Happy Camping!