Right accross from the box (Well a little west towards the river) there is plenty of parking and grassy areas. The parking lot can get very hot, but the grass fields are great.
There are over 1,000 large live oaks on campus. Try to park near one to take advantage of picnicing in the shade, but never park over the roots of the tree, you will be ticked and towed. Stay on the pavement. You won't have trouble finding the tailgaters near the Box. They start arriving the day before the game.