With the title "Parallel World", I just understand it like the author imagining another story with the same caracters. It doesn't make sens otherwise, well to me anyway!
From what I can tell, Kyosuke suffered an overdose from the drugs he was initially forced to take and then lost his memories. He has since tried to clean up his habit by going to rehab at their local hospital, but he has no real recollection of the events before. As for the parallel world extra, that's sensei's way of giving Ito-san and Kyosuke their happy ending. It takes place in a parallel world with the belief that there's an infinite amount possibilities that our lives have taken had we made different choices/or things had happened differently. In their world, Kyosuke's mother never gave him up and Ito-san was never kidnapped and groomed as an assassin. They're watching a movie about Kyosuke the prostitute and Ito-san the assassin on the TV, but at the very end the TV we are watching is of Kyosuke the student and Ito-san the teacher. Both worlds are equally real, and we can take solace in knowing that they are happy together in that world, at least. Even then, there's the underlying notion that no matter what world they inhabit, they will always manage to find one another.
Your welcome, hope my explanation wasn't too long winded ^^
Did they both forget one another, or was it just Kyosuke? Was the whole thing a dream, and their actual reality was that they were student and teacher? Wasn't he working in a shop or something... Someone help me out here! But all in all I did love the story, I love the darkness of it(▰˘◡˘▰)