According to this article, on Aug. 31, RL canceled previously made camping plans for two people from Sept. 1-3. It makes sense that she knew her son was coming home by Aug. 31. Then on Sept. 3, she reserved a spot for the three of them for Sept. 6-8.
So, while I think it's possible BL called his parents from the road and told them what happened, I have to entertain the idea that possibly he called them and told them he was coming home, but told them something else about GP. If they took a break after the Moab altercation, maybe his parents figured this was another break and had no idea about GP's death when he came home. If it's accurate that GP's mom didn't try to get ahold of them until the 9th or 10th, what's to say BL's parents didn't just think GP was with her folks, or whatever, for those 10 days. If BL made up a plausible story, they had no reason not to go about life, camping, buying a new phone, etc., none the wiser. Until GP's mom starts trying to communicate. Then, maybe his parents were like, "Hey, Bri, what's this? I thought you said she was at home?" (Or whatever...). If they questioned him and he finally fessed up, it was probably shock, figuring out what to do next, and calling the lawyer. IDK...