Preheat oven to 400 degrees. Heat olive oil in a large skillet over medium heat. Press a scant tablespoon parsley onto skin side of each breast. Season both sides with salt and pepper. Place breasts, skin side down, in skillet; cook until golden, about 8 minutes per side.
Remove chicken from skillet, and transfer to a roasting pan. Place roasting pan in oven. Roast until chicken cooks through, 15 to 20 minutes.
Meanwhile, return skillet to stove. Add onion; cook, stirring, until just golden, about 2 minutes. Add mushrooms; cook until soft, about 5 minutes. Add wine; scrape skillet's bottom to release cooked-on juices. Cook until wine is almost evaporated, about 5 minutes. Add vinegar; cook 1 minute more. Stir in remaining parsley; season with salt and pepper. Remove from heat; stir in butter.
Remove chicken from oven. Place spinach on plates. Slice each breast into sixths; arrange over spinach. Spoon mushroom mixture over chicken, drizzling juices over spinach, and serve.