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.

Due to some comments below, I see the need to mention that merely proving this is a parable is not sufficient to prove that the setting for the parable is purely fictional. It does not follow that if this is a parable, then nobody will experience torment in Hades. The use of this scenario in Jesus' teaching is evidence for its possible reality, although not as good evidence as its use in a direct teaching passage about all people rather than about one person.

Again, I hope we can discuss whether this is a parable without bringing external concerns into the discussion, such as whether it supports a preferred view.