Coming from Ft. Stockton, they would certainly have entered the park from the north, at the Persimmon Gap Visitor Center. If I'm recalling correctly (and it's been a few years), there is no kiosk/entrance station on the road, with a ranger inside.
You pull over into the visitor center parking lot to pay your required entrance fee inside, get maps, info, permits or just drive on by if you don't need any of those.
AFAIK, one does not need a permit to drive on the Dagger Flats Auto Trail or Old Ore Road. One does need a reservation to use one of the campsites along Old Ore Road.
If they stopped at the Persimmon Gap Visitor Center to pay their required entrance fee, there will be a record with the date and time they entered the park.
If they didn't stop in, that's another red flag.