March is my favorite month in San Antonio. It starts with the trees budding and in a week or so they are almost completely green. Wildflowers begin to show up in late March and although there may be some rain, it's not a high rain month. Danger of nighttime freezing has passed and daytime temps are generally mild, around 70.
For a park inside San Antonio proper I suggest Travelers World but my preference is something on the outer edges. Hidden Valley is 15 minutes from downtown but in a small town on the edge of SA...Von Ormy. Private fishing creek, country atmosphere and decent wifi. Attentive customer service and lots of help finding your way around the downtown area and finding parking and unusual places to visit. Really enjoyed their suggestion of the Toyota Factory Tour only a few miles from the park.
They have a gravel entrance road but not a problem for us.