Removal proceedings against me was terminated in my asylum case since June 2006.
Do I qualify for advance parole? If so what would be the maximum period of the reentry document? How do I answer the question "Why do I think that I am qualify for advance parole"?

Need much more specific information regarding the asylum case and the removal hearing. Best to consult with an experienced immigration attorney, it will be worth the consultation fee to be sure of your status on traveling.