If you're going to the west tower of TCH this is where you're headed, but the other buildings set up are very similar too--
https://www.google.com/maps/@29.7086946,-95.4016131,3a,75y,59.59h,88.86t/data=!3m6!1e1!3m4!1sUUkwNTTi5L6roc11FMsrvA!2e0!7i13312!8i6656
TCH is to the right, with low clearance underground parking pictured, or you pull around for valet (clearance for valet is OK)
The road is Fannin, with MetroRail restricted left turns in the middle as well as Metrorail platform in the middle, constant source of pedestrians, you have additional signage (that is unexplained) depicting when MetroRail is coming and when entering the left turn lanes is not allowed. No parallel/street parking in the heart of the medical center. As far as offsite parking/MetroRail one option is the Houston Zoo/Hermann Park nearby that has huge surface lots, but it's a hike in the heat even to get to the MetroRail stop.
If the TCH valet can park your car- excellent option. If not your best bet is probably park offsite and use the MetroRail or Uber.