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No need. Advanced parole don't serve here for your coming new status of holding valid H-1. Do prepare as many as work related documents with you when enter US custom with H-1B. They are things like business card and company badge if you have them already. Or for new hire, doc like hiring letter/offering letter. Everything you have with the new company who issues the H-1. US custom oftenly do ask you to prove if you are truely working with your H-1 sponsor.