That's one of the problems with Hollywood. It'll say 'policeman,' and an unless it says 'black policeman,' a lot of times you won't even get the opportunity to read for it, which is kind of crazy, but that's the way it is.
As far as being a deliveryman is concerned, I see nothing wrong with being a delivery man. You know, black or white or Hispanic or Chinese or whatever, you know? It's a job.