There are a couple of places to park. Both of them are close to the camping area. Parking attendants will try to get the vehicles close together not to waste parking space. If you tell them what you plan to do, they might try to accommodate you. Getting there early will help. It will be a little bit of a walk to your vehicle. You might find it's more convenient to just have your cooking close by. There is plenty of room to spread out at Lawson. It is the most spacious of all the TBF campgrounds.
On this Google map one of the parking lots is what shows as the Keystone Gorge Loop Trailhead and they also use what shows as the Lawson Hill Park and Ride. Camping is the big green open area next to the river. The tan area at the west end north of Telluride Crossfit is a baseball diamond but the lines don't show on this photo. That is the check-in and loading/unloading area.
https://www.google.com/maps/search/lawson+hill+camping+telluride/@37.9476389,-107.8771942,371m/data=!3m1!1e3?hl=en