The two delivery driver jobs that I had both paid per delivery. I was not paid to wait around. I would expect that if a delivery driver was paid per hour, that they would be given other tasks if deliveries were slow.
I've done a couple of food delivery jobs prior to college - as a pizza delivery driver, I was paid to be available (I made boxes, did light cleaning and tried to look busy while I was waiting for a delivery) and got consistent ($4) tips per delivery (not to mention all the pizza my midnight b-ball crowd could eat).
When delivering food for a small Chinese restaurant I was paid per delivery but there was a set run (usu. 4-6pm) and I got to expense miles, etc. No sitting around.