TRP has a small parking area and it'll be full of people camping. If you arrive the morning of the ride you won't find any parking there, they won't even let you in so don't try. But there's nice wide shoulders on the road for like two miles before TRP if coming from the north, i.e. Tahoe, and also some to the south as well. There's a lot of parking on the side of the road, you won't have a problem finding a spot. Usually the only other cars on the road are also Death Riders so when you get to where cars are stopping and parking, stop and park with everyone else. When you're ready to go, you may have to ride a bit to get to the official start. If you want, you can skip that part on the way back; you'll have done the full mileage.
There's a side road on the other side of the highway from TRP. Don't park there, you'll get a ticket or towed. Take a look at the road shoulder before you drive onto it; in a few places it's soft gravel and I have seen cars get stuck.
If you're stressed about finding a place to stay make reservations in south lake tahoe now.