Luke 16:19-31—Is the rich man in Hell or Purgatory? Luke 16:19-31 (NASB)
Now there was a rich man, and he habitually dressed in purple and fine linen, joyously living in splendor every day. And a poor man named Lazarus was laid at his gate, covered with sores, and longing to be fed with the crumbs which were falling from the rich man’s table; besides, even the dogs were coming and licking his sores. Now the poor man died and was carried away by the angels to Abraham’s bosom; and the rich man also died and was buried. In Hades he lifted up his eyes, being in torment, and saw Abraham far away and Lazarus in his bosom. And he cried out and said, ‘Father Abraham, have mercy on me, and send Lazarus so that he may dip the tip of his finger in water and cool off my tongue, for I am in agony in this flame.’ But Abraham said, ‘Child, remember that during your life you received your good things, and likewise Lazarus bad things; but now he is being comforted here, and you are in agony. And besides all this, between us and you there is a great chasm fixed, so that those who wish to come over from here to you will not be able, and that none may cross over from there to us.’ And he said, ‘Then I beg you, father, that you send him to my father’s house—for I have five brothers—in order that he may warn them, so that they will not also come to this place of torment.’ But Abraham said, ‘They have Moses and the Prophets; let them hear them.’ But he said, ‘No, father Abraham, but if someone goes to them from the dead, they will repent!’ But he said to him, ‘If they do not listen to Moses and the Prophets, they will not be persuaded even if someone rises from the dead.’”

Question: Could it be that the rich man was in Purgatory, and not Hell?
The rich man had compassion for his brothers. This does not seem likely if he were in Hell. If the rich man in Luke 16 is in Hell (and Lazarus in the bosom of Abraham, waiting for the Christ to release him to Heaven) why would he, the rich man,  try to keep his brothers from going to where he is?
In Hell they want everyone to die, not live, I believe. Could this mean that the rich  man was in Purgatory instead?
It's highly unlikely that the rich man is in Purgatory, as opposed to Hell, for at least a few reasons, as I see it:

1) This seems to be a parable about real persons, but related in parabolic form (characters in parables do not usually have names). So that seemingly mutually contradictory aspects therein are simply Jesus including all He needs to teach about the in the one simple story (i.e. Hebrews 9:27).

2) Jesus famously taught that it is extremely hard for anyone who clings to their riches or possessions to enter heaven (Lk 18:24-25). So that using a rich man as an example of someone who 'got off with it'1 would seem undesirable here.

3) "If they do not listen to Moses and the Prophets, they will not [believe]2 even if someone rises from the dead" describes an unforgiven sin—one of not believing unto salvation. This does not constitute a mere venial sin, but a damnable, that is, mortal one (cf. Matthew 18:17b).

4) The use of "father" and "son" doesn't necessarily mean amicability, especially because it's a parable, but also as neither does "friend" in Jesus' usage elsewhere.3

5) "between us and you there is a great chasm fixed" and "saw Abraham far away" would seem to put this place of torment out reach of any hope—"in order that he may warn them, so that they will not also come to this place of torment" seems to be the didactic focus of the parable, and so here Jesus eases out of the parable a little here towards the end and into the conclusion or point He wants to get across (as I find He does with others). In this case, the point He is getting across is the inability to change yours or others' situation once you die.



Footnotes

1 Purgatory isn't 'getting off with it' but as far as this rich man being in Purgatory instead of Hell, it would give the same, counterproductive impression here.

2 πείθω—to be persuaded or conviniced; by implication: believe (root of the word for 'faith' πίστις)

3 e.g. in Matthew 20:13; 22:12; 26:50 Jesus terms reprobates 'friend.'