HooMan:
Based on that link, it's clear that an H-1 holder can return on AP and continue working on H-1 status. The following paras mention that:
- An H-1 or L-1 holder who entered the United States on advance parole can apply for an extension of H or L status, and the approval of that extension would enable the person to return to the U.S. on H-1 or L-1 status. (To maintain that status, the person cannot seek outside employment.)
- An H-1 or L-1 holder who travels out of the United States, and returns on advance parole, is authorized to continue working for the H-1 or L-1 employer. He/she would not be required to obtain an EAD to work for this same employer, within the validity dates of the H-1 or L-1 petition approval.
- An H-1 or L-1 holder who has entered the U.S. with an advance parole document may depart and return as an H-1 or L-1 holder if that status has not expired.
Based on that link, it's clear that an H-1 holder can return on AP and continue working on H-1 status. The following paras mention that:
- An H-1 or L-1 holder who entered the United States on advance parole can apply for an extension of H or L status, and the approval of that extension would enable the person to return to the U.S. on H-1 or L-1 status. (To maintain that status, the person cannot seek outside employment.)
- An H-1 or L-1 holder who travels out of the United States, and returns on advance parole, is authorized to continue working for the H-1 or L-1 employer. He/she would not be required to obtain an EAD to work for this same employer, within the validity dates of the H-1 or L-1 petition approval.
- An H-1 or L-1 holder who has entered the U.S. with an advance parole document may depart and return as an H-1 or L-1 holder if that status has not expired.