With few exceptions, leaving the United States while a green card case is pending can be fatal to one's case. In general, departure from the United States without an advance parole document normally results in a denial of the adjustment of status petition. USCIS will determine that the individual has abandoned the application. Moreover, in some cases, if an applicant has overstayed and accrued too much "unlawful presence," he or she may trigger a three or ten-year bar from returning (depending on the length of the unlawful presence). So, in most cases, if an individual foresees that he/she will need to travel outside the US during the pendency of an adjustment of status case, advance parole should be strongly considered.
The best rule of thumb regarding travel during the pendency of your green card case is not to do so. We have seen a number of individuals unknowingly sabotage their adjustment of status applications by traveling outside the US without taking the proper precautions. The concept is this: when an adjustment of status application is filed with USCIS, the individual is applying to convert his/her status to that of a permanent resident inside the United States. With a few limited exceptions, any trip taken outside the US while the adjustment is pending is normally construed as abandonment of the application. Therefore, it is extremely important that if you have an I-485 pending with USCIS, you remain in the US. Any departure may result in not only a denial of the adjustment application but also potentially obstruct readmission. Worse, a departure may trigger an unlawful presence bar (either three or ten years, depending on the length of time an individual has been unlawfully present) and prevent the individual from re-entering.
Advance parole is most often understood in the context of adjustment of status applications where the applicant is given advance permission by USCIS to leave the US while his or her green card case is pending without being considered to abandon it. However, one does not necessarily need to have filed an application for adjustment of status (I-485) in order to apply for advance parole. Individuals who have been granted DACA status, for example, are also eligible, under certain circumstances, to apply for travel permission.