One thing that seems to distinguish parables from accounts of actual events is the absence of specific names for the people in the parables. In the parables we read, "A sower" (Luke 8), "A rich man" (Luke 12, 15), "A man" (Luke 13, 14), "A Samaritan" (Luke 10).
However, in the account of the Rich Man and the Beggar, we are actually given the name of the beggar. This is quite a distinction from all the other parables. In addition, Jesus tells us that the rich man saw another specific man--the patriarch Abraham.
So, the fact that the rich man's name is withheld does support the idea of this being a parable. Yet, the fact that two other specific people are identified by name seems to be stronger evidence to support the idea that this was an actual historical event with actual people.
The significance of this story comes in the reference to someone rising from the dead, as Jesus states that even if someone rises from the dead, some will not believe. This is fulfilled in the fact that Jesus rose from the dead, and even the chief priests of the day continued in their refusal to believe in Him.
Narnian's answer covers well the argument for this story being real and not a parable. I will attempt to cover the argument that it is a parable.
The first thing to note is the parable's location among other parables and teachings. Luke 16, where the parable is found, is surrounded by other parables and teachings. It is not part of the historic sections of Luke. It also sounds like a parable. There's a point to the story that has very little to do with the actual story. This alone is evidence enough for me to call it a parable, but there is more.
The only argument I have seen that this is a real story and not a parable is based on the fact that specific names are mentioned. I personally do not find that convincing.
Yes, it is odd that there are specific names in this parable. The appearance of Abraham is easy to explain. He is a character of necessity because of the place where Lazarus is. Many translations call it "Abraham's side" or "Abraham's bosom".
This is the name of a place, not a person. Abraham has become a necessary part of the story. He is the voice that will deliver the message of the parable. There is little significance that it is Abraham rather than someone else.
Lazarus, on the other had, doesn't seem to be needed in the story, so the name was mentioned for a reason.1
One cannot read the parable of Lazarus and the Rich man and not be reminded of the resurrection of Lazarus in John 11. When examining that story, we see some similarities, especially in the message that can be gleaned from it compared to the message given in the parable.
At the end of the raising of Lazarus we do not see all the people falling to worship Jesus. Instead we see a few are convinced then a plot to kill him quickly develops.
But beyond all this is that parables are meant to make a point. They are meant to illustrate a Truth, a bit of wisdom, that can easily be separated from the story. It is plainly clear what the point of the parable of Lazarus and the Rich Man is:
In light of
which sounds oddly like the message of the parable, it seems plausible that Jesus developed this parable based off of the reactions of the pharisees after Jesus really did raise Lazarus. Compare this to Who are the 5 Brothers Mentioned at Luke 16:28 - Rich Man and Lazarus, where my answer discusses the argument that they are the brothers of Caiaphas. This makes for a pretty good argument that Jesus made this parable drawing from real events, and the mention of Lazarus and 'five brothers' was a subtle way to name the Pharisees as those who would not believe "even if the dead returned."
An illuminating Source - Lazarus in John and Luke - Crisis Magazine - MARK P. SHEA
The story of the rich man and Lazarus appears in Luke's Gospel as part of a group of parables, because of which it is unlikely to have been a true event. More importantly, it has close parallels to the rabbinic parable of Bar Majan, which seems to have been the source for the Christian parable. If it is possible the Jewish parable is no older than Luke's Gospel, it would be technically possible that the Bar Majan story was derived from Luke but unlikely, whereas the gospels made much use of Jewish traditions. The Jewish parable is said to have evolved from the ancient Egyptian story of El-Azar
Douglas Welker Kennard (Messiah Jesus, page 93) says this parable is similar to the Jewish parable of a rich tax collector named Bar Majan and a poor teacher of Law who reversed fortunes in the afterlife. Kacy Madsen ('The Rich Man and Lazarus') summarises the story of Bar Majan and its Egyptian source, and says that both Jesus and the Pharisees would have been familiar with this folklore.
As a footnote to this story, the New American Bible (NAB) says:
This is a real story. Jesus started out by saying, "A certain man". And he gave a name, Lazarus. Parables don't have names, it's mostly servant, or vine dresser etc. and a timeline, "they have Moses and the prophets". And why didn't the rich man had a name? Jesus said ..."depart from Me, I never knew you".