One more point in favor of the story being a parable, aside from those above, is that it contains internal self-interpretations which are unlikely in a straightforward recounting, but perfectly sensible as one of Jesus' self-interpreting parables (other examples are the Wedding Feast and the King's Far Journey, but the other parables in Luke 15-16 are also mostly self-interpreting).

The two most obvious example of self-interpretation are 

 1. when Abraham describes an abyss which the rich man should have been
    able to see,
 2. when Abraham ends the story by warning that stories of the afterlife
    won't be effective in making people repent.

That last is especially convincing to me: if that had been a real story giving real details, then there is no reason to end it with the warning that giving real details won't help.