There is no membership which allows people special access to reserve
specific campsites prior to arriving at a park. All Texas State Park campsite reservations are made through the toll-free state reservation line in Austin or online, and are for a TYPE of campsite, not a specific campsite. For example, "1 campsite with water and electricity in Campground A". However, most Texas State Parks allow reserved guests arriving at the park to drive around the camping area they've reserved a site in to locate a specific campsite they would like to stay at (if it's available). That does not mean the campers were allowed to reserve a specific campsite prior to arriving.
The busier parks do post notes or "Reserved" signs on their campsites to help keep track of campground occupancy and help communicate to arriving reserved guests which sites are not available. I've seen this done at parks such as Inks Lake State Park and perhaps Garner State Park -- two of the busiest parks in Texas. However, I think if you asked the rangers you'd find there are no specific names tied to those campsite reservations -- the signs are just noting that yet another site in the campground has been reserved. The exception to this would be things like screened shelters or cabins, which might be reserved ahead of time to a specific person or party. If you do see names, it may be a case of reservation-holding campers who have already checked in but who are out enjoying the park rather than parked and set up yet at their site.
If anyone refers to a "special pass", they're more than likely referring to the annual
Texas State Parks Pass (formerly called the Texas Conservation Passport or "TCP"). The pass exempts passholders from the day use entry fees to Texas State Parks for 1 year, but does not give them privileged access to reserve specific campsites prior to arrival. There are some other park passes, but they're similar in features and are offered to retirees and other specific groups. The Texas State Parks Pass is available to anyone who pays the annual registration fee ($60).
I hope this helps.