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
- when Abraham describes an abyss which the rich man should have been able to see,
- 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.