The reason UT would be able to redevelop Centennial Park is because UT owns that land. It's not city owned land, so it's not a city park. UT could build a Walmart there if they wanted to.
The zip code for Centennial Park is 78701. As you can see from this map, the park isn't listed as a city owned park. Neither is Clark Field, which is also owned by UT.
http://www.austinparks.org/our-parks.html
Also I wouldn't be surprised if after Brackenridge is replaced we saw it torn down, and the land turned into more park space adjoining Waterloo Park to the west. Or they could at least realign Red River so that it would cut straight through the Brackenridge property just east of the parking garage. That would allow the land where the Brackenridge garage is now to join the rest of Waterloo Park. It would also move Red River Street away from Waller Creek and have it be a more intimate park.
And actually, straightening Red River would ease traffic congestion. I don't see how realigning it would cause more traffic problems. Curvy streets usually lead to traffic congestion.