TexasCHLforum.com

My understanding of the law:

A privately owned parking lot can be posted with a 30.06 sign. A publicly owned one cannot.

The premises definition is for things like schools where you can't carry into the building, but can lock the weapon up in the parking lot or carry concealed when dropping little Johnny off.

From the LS-16 FAQ

Q: If I drive to a shopping mall that does not permit handguns, will I be
allowed to park in the parking lot and leave my gun in the car?

A: If the parking lot is not posted with the sign described above (30.06),
handguns may be left in the cars. However, if the sign is posted in or
at the entrance to the parking lot, you may not.

Q: Can a license holder have a handgun in the license holder's
vehicle in a school parking lot?
A: A license holder is not prohibited from having a handgun in his or
her vehicle in a school parking lot. (School employees should know
and comply with their employer's policy on this point.) However, it is a
criminal offense for any person who is on school property to exhibi,
use, or threaten to exhibit or use a firearm. See Texas Education
Code 5 37.125(a).