I've doubted that he flew home, and the reasoning just didn't make sense. Also, in my experience, when parents don't care for a boyfriend or a girlfriend, as it seems Gabby's dad felt about Brian (Brianne), I think kids are more likely to lie to keep their parents from being angry. JMOO, but I can imagine Gabby's dad not wanting her to stay in the van by herself if Brian was gone a few days, so he might have sprung for the hotel room. And, the pizza one day.
Something like that--getting off the road and into a comfy hotel--could have been all Brian's idea, and he convinced Gabby to lie to her parents. I imagine van life would get pretty old, pretty quickly, if you didn't really love it.
Knowing Gabby's mom said the information was "mistaken," leads me to believe she originally was told BL flew home, but when the FBI started looking into it, it was probably easy to determine he hadn't flown at all.