Glue a metal rail with double-sided tape on the parquet, so you don't have to screw it in if you want to remove it sometime in the future and frame up the wall. You shouldn't put plastic in the wall when it's indoors, and never double plastic in any wall (as this traps moisture).
The part against the door can be soundproofed with a 50 mm foam board instead, so you don't need to put plywood behind it. OSB works behind drywall if you want to save some money...