You should be able to find public metered parking along the streets near the capitol. There is a Capitol Visitors Parking Garage 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: $8.00).
Parking is also available at the Bob Bullock Texas History Museum underground parking garage for a fee of $8. The museum parking garage entrance is located on the south side of the Museum on 18th Street, just off North Congress Avenue.