Trim the fat from the breasts and remove the white tendons running through the tenderloins. Place each breast between sheets of wax paper and gently pound with a meat mallet until less than 1/4 inch thick. Dust the chicken with the flour until well coated on all sides. Sprinkle with salt and pepper.
Heat the oil in a large skillet over medium-high heat until very hot but not smoking. Use the tongs to add the chicken breasts in a single layer and saute for 3 to 4 minutes on each side, until firm. Remove to a serving platter and cover with aluminum foil to keep warm.
Add the mushrooms to the skillet and saute until the mushrooms have given up their juice, about 8 minutes. Stir in the parsley. Season to taste witih salt and pepper. Remove from the heat and whisk in the butter, 1 T at a time.
Pour the sauce over the chicken and serve immediately.