There is a main parking lot outside of the Main Gate. The website has more information if you need specific directions. Parking is free and first-come-first-serve.
You can access Barton Springs Pool through either Zilker Park or enter on the backside via the Robert E Lee Parkway
There is plenty of parking within the park. Depending on your location within the park.
From Barton Springs Road turn left into Andrew Zilker Road where there are parking lots on either side of the road. Then it's just a short stroll downhill to the entrance. We visited 10 days ago; there are now parking meters with $5 charge. As mentioned in my review, get there by about 10am on a weekend, both for parking and quick entry.
Get quick answers from Barton Springs Pool staff and past visitors.