Two fundamental rules of good biblical scholarship will help us to understand a difficult scripture almost every time. First, look at the immediate context. Second, look at the broader context of the entire Bible. By contrast, attempting to understand a difficult passage by human reasoning is bad scholarship. Let's apply the proper rules systematically to the parable.
The immediate context (the verses before and after the parable) doesn't clearly provide us with further information, so let's go on to the second rule. Jesus said, "The Scripture cannot be broken" (John 10:35). He meant that every section of the Bible on any given subject must be in agreement with every other section that addresses the same topic.
What does the Bible say about the state of the dead? A single phrase in Ecclesiastes 9:5 summarizes the Bible's teaching about the state of the dead: "The dead know nothing." (See also Ezekiel 18:20; 1 Corinthians 15:22-23, 51-52; John 3:13, 16; Acts 2:29, 34; Romans 6:23 and many other passages that show that the soul can die, that physical death is like sleep and that the dead will be resurrected in the future, that no one has gone to heaven, that sin leads to death—not eternal life in another place, etc.).
Reading this parable to mean that the dead are conscious in an ever-burning hell or in heaven would be a clear contradiction with the broader context of the entire Bible. Indeed, it would be a misinterpretation.
However, we must understand that this is a parable—"an imaginary story…to illustrate and inculcate some higher spiritual truth" (International Standard Bible Encyclopedia, "parable"). If we take its lesson at face value, it is in complete harmony with the rest of the Bible. Jesus' straightforward point is that it is too late to change one's behavior and character after death. His point is that we must live in a godly manner when we are alive, just as the man Lazarus did in the story.
We bring together all of the scriptures about the afterlife in our booklets Heaven and Hell: What Does the Bible Really Teach? and What Happens After Death? They also contain further background information about this parable.