Jesus does not say the rich man is taken immediately to an eternally burning hell. He says the rich man dies and is buried. People are buried in a grave and covered with earth. Hades (verse 23) is the Greek word for "grave." The King James Version generically translates hades into "hell," as it also does the Greek words tartarus (the present condition of darkness and restraint of the fallen angels or demons) and gehenna (a place at the bottom of a high ledge at the south end of Jerusalem where garbage and dead bodies were dumped and burned). Other Bible translations correctly distinguish the different meaning in these words. The rich man went to the same kind of place Jesus did when He died—"hell" (KJV) or "Hades" (NKJV)—but the Father did not leave Him there (Acts 2:31-32).
Daniel 12:2 speaks of those who will be resurrected to eternal life (the just) and of those who will be resurrected to damnation or judgment (the unjust). In the parable, Jesus speaks of two different, separate resurrections (John 5:28-29; Acts 24:15; Revelation 20:4-5, 11-12). Jesus pictures the rich man as wicked and lost, but even he will open his eyes and rise from his grave after the Millennium. Having passed up his opportunity for immortality by choosing this world's temporary, material riches and pleasures rather than eternal, spiritual riches, he is without hope, doomed to perish in the Lake of Fire.
The Parable of Lazarus and the Rich Man shows the resurrection from the dead, not an instantaneous going to heaven or hell. It is a resurrection from death, not from life. It depicts mortals who die and are dead, not immortals who never lose consciousness and live forever under punishment in a fiery hell. Jesus describes bringing back to life one who was dead, who had no conscious realization of the lapse of centuries and millennia since his death.