I did this hike earlier this year. You will have fun. Our group dropped off our car on the South Rim and did the shuttle over to the North Rim. To start our hike,we got dropped off by the Lodge shuttle. I don't remember any signs saying no overnight parking, but since we were not having to park there, I could have missed it.
I have a friend that went rim to rim to rim the same day as our hike and I will send him an e-mail to see if he knows.If he doesn't, he is going back on Oct. 8th for another r2r2r (crazy) I will ask him to check it out if that is soon enough for you.
