I don't think that Jesus would have used the parable of the rich man and Lazarus if parts of it were untrue. The rich man's soul was not asleep-he was in torment. Lazarus was not asleep, but rested in Abraham's bosom. Jesus also stated to the disciples that Lazarus was asleep, when he was dead. I believe by that Jesus was stating that his soul entered into a place of rest. Sleep was not a euphamism for death at that time, because the disciples didn't understand-thought He was speaking literally. Even in the Old Testament,
I Sam 28:15 And Samuel said to Saul, Why hast thou disquieted me, to bring me up? would indicate that Samuel was at rest after his death. Part of this "rest" may be that we do not know all that is happening of evil in the world... Lazarus didn't answer the rich man-Abraham did. But that passage may also indicate the rich man's pride, in that he would speak to Abraham but not to Lazarus even in death.
For people to go straight to Heaven (where God's throne is) or Hell (where the devil will be) upon death seems odd to me, since at the end,
Rev 20:12 And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works.
So I don't really have an answer, but I believe that at least some people will be at rest after their deaths, not asleep necessarily, but at peace from the troubles they faced while alive.