Preheat the oven to 400 degrees F. Line a rimmed baking sheet with parchment paper.
Stir together the parsley, olive oil, smoked paprika and garlic in a small bowl. Lay the puff pastry sheets flat on a work surface and cut one sheet in half; reserve one of the halves for another use. Rub the parsley mixture evenly over the 1 1/2 pastry sheets. Cut the one whole pastry sheet crosswise into thirds, then cut each third crosswise into 1 1/2-inch-wide strips. Cut the half sheet crosswise into 1 1/2-inch-wide strips, as well.
Place a piece of chorizo on the end of one dough strip and roll the chorizo in the dough as you would a cinnamon bun. Press firmly at the seam to seal, then place the chorizo roll, seam-side down, on the prepared baking sheet. Repeat with the remaining chorizo and dough strips, leaving about 1 1/2 inches between each chorizo roll on the baking sheet (you will have a few extra dough strips).
Bake the chorizo rolls until the pastry is puffed and golden brown, 15 to 20 minutes. Transfer to a platter and serve warm.