Doesn't this parable indicate that people have some kind of soul or spirit that is conscious even after death?
Luke 16:19-31 is perhaps a difficulty for proponents of the soul sleep theory, but it is not an outright contradiction of the theory for a number of reasons.
Scripture, in both old and new testament, makes frequent use of literary personification. The 'blood of Abel' cries out in Gen 4:10. A prophecy in Ezek 32 uses the visual of leaders speaking from the realm of the dead (Ezek 32:21.) Floods clap their hands and hills sing together for joy in Psalm 98:8. Thistles talk in II Chron 25:18, etc. A parable in Judges 9 begins, "Once upon a time the trees decided to choose a king. First they said to the olive tree, 'Be our king!' But the olive tree said to them, 'Shall I leave my fatness with which God and men are honored, and go to wave over the trees?'…'"
While the use of personification in some places does not mandate that it is being used in all possible places, that it can be used is important to keep in mind when facing scriptures that seem to 'contradict' on the surface. Just as the tree parable shows trees discussing a monarchy (when all would agree that trees do not speak) it is possible that the Parable of Lazarus and the Rich man show the dead speaking figuratively, not literally.
Proponents of soul sleep will likely point to many verses on death that seem to concretely say that the dead have no thought or consciousness:
"For the living know they will die; but the dead do not know anything, nor have they any longer a reward, for their memory is forgotten." Ecc 9:5
"And many of them that sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt." Dan 12:2
John 5:28-29 "...An hour is coming, in which all who are in the tombs will hear His voice, and will come forth; those who did the good deeds to a resurrection of life, those who committed the evil deeds to a resurrection of judgment."
Etc. Any theory (not just soul sleep) is faced with reconciling verses that seem to imply that there is no conscious thought in death with verses that imply the spirit is conscious.
No one verse or passage, such as Luke 16:19-31, is going to 'prove' the case definitively one way or another on its own.
Another thing to keep in mind is that the story of Lazarus and the Rich man is a parable - a moral story using elements from life or culture that would be readily understood on the surface, but hid a deeper spiritual meaning. With parables, the story itself was not necessarily always based on a true occurrence. Did a good Samaritan help a man beset by thieves? Did a specific woman really lose one of her coins? We don't know.
The main spiritual point of the Lazarus parable is that there is no second chance for salvation after death, and that some won't be convinced even if someone rises from the dead (like Lazarus or Christ). The state of the dead is a setting - but not the main spiritual point. As such, it is hard to be 100% dogmatic that the setting is speaking of an actual occurence vs. pulling from imagery that would be understandable to the culture.
Beyond the possible use of personification and the literary style of the parable, there is a third reason that the parable of Lazarus and the Rich man would not directly contradict the theory of soul sleep. Note that the rich man is in 'Hades,' but is in fiery torment. Yet usually,'Gehenna' is used in the new testament when describing fiery torment (such as Mk 9:47-48.) Hades is equivalent to the OT term 'Sheol' and simply means the grave, or abstractly, the realm of the dead. Yet here, in this one instance, Hades is described as a place of fire.
This discrepancy seems unusual, but not perhaps in light of Rev 20:14, "Then death and Hades were thrown into the lake of fire. The lake of fire is the second death."
Hades will be thrown into the Lake of Fire at the judgment. As such, it is possible, even probable, that the Parable is presenting Lazarus and the Rich man as having already faced the judgement. (Lk 16:26 also hints at the two having already been judged.)
This would not conflict with the soul sleep theory and actually supports it.
[My personal opinion is that both the soul sleep and concious-between-death-and-judgment theory have seeming support from scripture as well as difficult passages to reconcile. However, Luke 16:19-31 is not a direct impediment to the soul sleep theory.]
The parable would be taken as allegory.
The Scriptures taken as a whole (tota scriptura) are understood by many to be clear that the bible teaches that "dust we are and dust we shall return".
It was the devil that suggested that Adam and Eve would not die when God was clear that they would.
The parable of the rich man and Lazarus surely should not contradict the plethora of scriptural verses which seem to point to soul sleep before the resurrection to come.
Read 1 Thessalonians 4:13 and the surrounding context...how can this be understood if we go straight to another realm upon death? With this and other verses which offer clear defence of death means death (not life somewhere else) should this not carve our understanding of those other more difficult verses such as the one you asked about.
Great question& God bless!