I traveled on AP via QR airways too- the ticket agents are aware of the documents and so are people in the immigration at TIA. Before you board your flight to the USA from DOHA , they do a document check and people there are also aware of AP. When you land in USA you will be taken to secondary inspection that's where they keep Paroled stamp. The waiting time varies from 2 mins to an hour. Mine was less than 5 mins. Officers might ask you questions about your trip but sometimes they don't so depend on the officer and POE.

Enjoy your flight!