Abraham told the Rich man that his state was fixed and that there was a great chasm between them and that no one could cross over to the other side. The story in Luke 16:23-31 is not a parable that Jesus was telling. Parables do not mention specific names of real people like Abraham and Moses. Lazarus died and went to Paradise, where Abraham gave him a greeting, a hug and the Rich man was in Hades. Hades is not an abstract location, it is a place where unbelievers go. To say that Purgatory exists is to say that Jesus work on the Cross was not sufficient, that somehow more needs to be done. He died for our sins, past, present and future and there is nothing more, (by works or prayers) that can be done once a person is dead. They are either with the LORD or NOT. Jesus never taught about Purgatory. Jesus said on the Cross, "It is finished!" Then He died and rose again on the third day. He told the person next to him that he would be in Paradise TODAY. This sinner repented and believed in Christ. That was enough. He did not go to Purgatory, He went to be with Christ that day.
Prayers for the dead comfort the living more than anything else. Prayers are talking to God and so if your loved one is in heaven, Jesus will pass on your message to them. But those in Hades ... have no comfort, no hope, they are lost and awaiting destruction.