Parking will be harder to find the closer you get to the capitol.There is a UT football game on October 29, so parking will fill up early depending on the start time of the game.
If you arrive early in the day you may 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: $10.00, however on special events on weekends the rate goes up, so expect to pay more on Oct 29).
Parking is also available at the Bob Bullock Texas History Museum underground parking garage for a fee. The museum parking garage entrance is located on the south side of the Museum on 18th Street, just off North Congress Avenue.