Again, for sake of argument, let's say BL lied to his parents and sister and they did not know anything was amiss until Sept. 10th, when GP's mom started texting the parents.
This would explain the camping trip and other "normal" behaviors witnessed by neighbors prior to that date. It would also explain BL's sister's GMA interview on the 15th, with her speaking in terms of time since she found out GP was missing, and her brother then missing. From Sept. 10th on, not including seeing him on the 1st and 6th because that was before she knew GP was missing. (Although my one question there is did he have and did she see the white van when he visited her on Sept. 1).
So the parents get a lawyer on or around Sept. 10, because they either finally know, or suspect what their son did, and then don't know how to proceed appropriately. Lawyer tells them to stay quiet and let him handle the press and LE, so they do.
With pressure coming from all sides now, Brian leaves and tells his parents he is going to the Carleton Reserve on Sept. 14. On the 17th, after BL not returning home, they find an empty gun case in the Mustang, note his new cell phone and wallet are home, and they think maybe he's gone off to harm himself. So they tell LE.
Maybe I'm giving his family too much benefit of doubt, but imo, I have to at least consider the possibility that they may not be lying or covering for him, at all. Maybe they really don't know where he is, are concerned for his safety, didn't know where GP was before she was found, and didn't know anything happened before Sept. 10th.