How can I travel with an H-1b visa and a pending advance parole application?

If you file an I-485 application to adjust, it is advisable that you should file both I-765 and I-131 so that you can obtain both EAD and Advance Parole (i.e. EAD/AP combo) document. The fact is you can use the EAD for employment and AP for travel purposes. Furthermore, your AP guarantees that you can travel out of the country without losing/abandoning your I-485 because, if you travel out of the country without a parole document in place, you may be unwittingly abandoning your adjustment application. Thus, never travel with the H-1b document once an I-485 is filed you must have and use AP document.

Yes, you have the option to travel under the valid/unexpired H-1B visa while advance parole is pending, without jeopardizing your I-485 or advance parole application.

You may travel with the H-1B with no problem provided you have the H visa to return. If you file the advance parole and travel before it is adjudicated, it is denied.

If your I-485 application and your I-131 application are pending, then you can use your H-1B visa to be readmitted to the United States.

If you have a valid H-1B visa in your passport, you do not need valid advance parole to travel while your adjustment is pending. However, if you file an advance parole application, and travel with the H-1B before the advance parole is approved, the advance parole application will be denied.

H-1B is dual intent, so you should be able to keep using it even while waiting on the I-485.

H and L adjustment of status applicants may travel on their valid H or L visas without abandoning their I-485 adjustment of status applications. All other adjustment applicants must wait for their advance paroles or they will abandon their applications upon leaving the United States. You should check with an experienced immigration attorney before making plans to travel outside of the United States.