Make copies
If you have valid H visa, then the airline agent may not ask for AP.If not they will verify the AP. In my case ( I flew KLM ) they did not take any copies, but I have seen postings where people have given copies.
At POE, they will take both the originals and give you back one copy with Parole entry stamp.
At Newark, we were ask to go to secondary immigration room , no questions were asked , everything was ok-ed in 10 mins.
All the best