Parking will be harder to find the closer you get to the capitol.There is a UT football game on October 24, so parking will fill up early depending on the start time of the game. It may be easier if you park further out and use Lyft or Uber to get a ride to the capitol. You may be able to get $20 in credit towards a free ride if you use the following links to get an invite: invite for Uber or invite for Lyft. The $20 credit is only valid for first-time users of either service. Sign up at the link, download either or both apps. It is very convenient and they usually pick you up in less than 5 minutes.
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 24).
Parking is also available at the Bob Bullock Texas History Museum underground parking garage for a fee of $20 on October 24. The museum parking garage entrance is located on the south side of the Museum on 18th Street, just off North Congress Avenue.