What are some specific examples in the movie O Brother Where Art Thou that parallel the The Odyssey?
One strong parallel between The Odyssey and the movie O Brother Where Art Thou is the blind railroad man who tells the three runaways their future (they will see a...a cow on the roof of a cotton house). In the Odyssey, Odysseus meets Tireseus from the underworld, also a blind man who tells him his future.
In O Brother Where Art Thou, the three men (Pete, Delmar, and Everett) meet three women who get them drunk and take Pete back to prison for money. In The Odyssey, Odysseus runs into the three sirens, beautiful women who sing and cause him to become confused. (In the movie, their song is "Go to sleep you little baby...").
Both Everett and Odysseus are trying to return to their families after an extended amount of time apart. Everett was in prison, however, Odysseus was nobly serving in the war. However, one difference between the two is Odysseus is a very physical, strong man while Everett is not as much physical as he is intelligent. However, when it comes down to it, they are both just men--compared to stories where the main character is some sort of demi-god (Hercules).
In the movie, Everett and Delmar meet Big Man Teague who effectively robs them under the pretense of wanting to begin a business arrangement with them. Big Man Teague also has an eye patch over one eye, causing him to look somewhat like a cyclops. Similarly, Odysseus has to run away from the brute of a cyclops--the difference here being that the cyclops is purely out for a physical destruction of Odysseus--Teague doesn't care if he physically destroys Everett and Delmar, but he cares about the financial side of it.