There is a rest area a few miles east of El Paso on I 10. It's about a mile and a half east of the overpass at the Fabens exit. No really level spaces in the parking lot on the westbound side but the east side has some ok places.
At exit 37, Horizon Blvd there are 2 truck stops on the north side, Loves and American, and Petro on the south side.
At exit 0 I 10 is going almost due north. There are a Petro and a McDonalds on the east side. The westbound exit was closed for construction, but I haven't been over there for a couple of months so it may be open. Just into NM westbound is the visitor center which has a good parking lot.
Would it be shorter to go up to I-40?
