One of the best forms of parody is to take a classic work and update it to the modern day. While an update is not necessarily a parody, intentional comedic elements can be introduced -- such as having the miserable pawnbroker, Alyona, being recast as a politician in Washington D.C. -- to make fun of both the original work and events in the modern day.
"Crime and Punishment" ends with Raskolnikov breaking down and confessing his crimes to Porfiry, the detective who was chasing after him. In the end, Raskolnikov is sentenced to a prison term in Siberia, where Sonia, a kind-hearted prostitute, follows him in hopes of giving him a chance at salvation; Raskolnikov has a ray of hope at the end of the work. A parody of this ending could have Raskolnikov simply get away with his crime and instead of having a guilty conscience, he is instead nonchalant about the murder.
"Crime and Punishment" is written from the third-person perspective. One way to parody this is to write it from the first-person perspective, with the characters' traits being accentuated. For example, Raskolnikov spends most of the novel fretting about the murder; his first-person perspective could involve a comic overindulgence in his neurotic nature. Additionally, there are several secondary characters in the novel, such as the landlady Praskovya Pavlovna. A short piece from her perspective could involve her wondering about how strange Raskolnikov seems.
A sequel to "Crime and Punishment" is also ripe with comedic possibilities. Perhaps Raskolinkov returns from Siberia and decides he wants to give another whack at murdering; maybe it could be something as short as they freeze to death in Siberia. Another possibility is a short prologue in which the other characters, like Raskolnikov's mother and sister, live out happy lives and secretly admit that they are happy that Raskolnikov is gone.