An Advance Parole (AP) document is issued by the United States Citizenship and Immigration Services (USCIS) and provides a non-immigrant K-1 visa holder permission to re-enter the United States while their Legal Permanent Resident Adjustment of Status (AOS) application is pending. Failure to obtain advance parole BEFORE departing the U.S. effectively invalidates your LPR application. It is not enough to have applied for advance parole, you must have received approval before you travel. See  When is Advance Parole Required For Travel?
You may apply for Advance Parole if you have an adjustment of emergent personal or bona fide business reasons. Due to the processing required to issue an AP it is advisable to apply far in advance of when you plan to travel. As of late 2007 AP's were taking up to 100 days to be issued. In an emergency a person may make an InfoPass Appointment at a local USCIS office and request an AP expedite. Proof of the emergency must be provided, along with a copy of the AOS NOA receipt.
IMPORTANT: Please note that expedited processing is done at the discretion of the Field Office based upon the evidence provided by the applicant. Approvals vary from office to office, as well as officer to officer, so what is standard at one office may not necessarily be standard at another. Example Advance Parole Form
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NOTE: The above information does not address the specific requirements for any given case and is not a substitute for the advice of an attorney.