You may want to park at the Capitol Visitors Parking Garage, east of the Capitol, located at 1201 San Jacinto located between Trinity and San Jacinto Streets at 12th and 13th Streets. Parking is free for the first two hours and $1.00 for each half hour thereafter (maximum daily charge: $12.00). If you go to the top floor and park near the elevator (there are stairs at each corner with glass doors; the elevator has concrete walls), you can take the elevator down to the 2nd floor, turn right, and it’s just a one-block walk to the Capitol grounds.
On busy days this garage may be full. There is metered parking (requiring coins), with time limits. But there are other garages in walking distance that may be a better option.