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I am not sure what you are referring to copy. When you get advance parole you get three originals from INS. If you return to USA the first time after you got your advance parole, the officer at the Indian airport should have taken one original leaving with the other two. Sometimes at intermediate airports such as Frankfurt, the officer takes the other original leaving with just one. This is all you need. The immigration officer at the port of entry will just look at this one original and stamp it. And you can use that same original as many times to re-enter into USA until the time of expiry of the parole. I hope this is what you are asking for and this clarifies your question.
I don't think "Advance Parole" is stamped as a visa. I am assuming that you are talking about your I-94 attached to you passport. If that is the case, yes you need the I-94 along with the original I referred to in the previous paragraph, to re-enter.