Heat the oil in a medium skillet over medium heat. Add the sausage and cook, crumbling it into small pieces with a wooden spoon, until no longer pink, about 4 minutes. Add the oregano and onion and cook until the onion is softened, about 6 minutes.
Make a space in the center of the skillet and add the tomato paste. Toast it for a minute, then stir it into the sausage mixture. Add 1/4 cup water and bring to a simmer; cook until the mixture thickens, about 2 minutes. Scrape into a medium bowl and let cool, about 15 minutes.
Meanwhile, preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper.
Stir the Parmesan into the cooled sausage mixture. Season with salt if necessary.
Unroll the crescent dough and separate into 4 rectangles (each made of 2 crescent dough pieces). Press the seams together to seal and create 4 solid rectangles. Roll or press each rectangle to about 6 inches by 8 inches.
Lay out one of the dough rectangles with the long sides on the top and bottom. Layer a quarter of the spinach leaves on the dough, leaving a 1/2-inch space at the top and bottom but extending the spinach to both sides. Sprinkle with a quarter of the parsley. Spread a quarter of the sausage mixture over top. Sprinkle with a quarter of the mozzarella. Press down on the cheese to compress the filling and adhere it to the dough. Roll the dough tightly over the filling and pinch to seal the seam. With a sharp knife, cut into 6 rounds. Place the rounds cut-side up on one of the prepared baking sheets. Repeat with the remaining ingredients to make 24 pinwheels.
Bake, rotating the baking sheets from top to bottom halfway through, until the dough is puffed and browned and the cheese is browned, 12 to 14 minutes. Serve warm.