Pat chicken dry and season with salt and pepper. In a heavy skillet heat oil and 1 1/2 Tablespoons butter over moderately high heat until hot but not smoking and brown chicken in 2 batches, transferring with tongs to a large plate as browned. Discard all but 1 T. fat from skillet and saute onion and mushrooms, stirring occasionally, until liquid mushrooms give off is evaporated. Add Marsala and cook mixture until Marsala is almost evaporated. Add broth and chicken with any juices on the plate and simmer, turning chicken once, until cooked through, about 15 minutes. Transfer chicken with tongs to a platter. Simmer mushroom sauce until liquid is reduced to about 1/2 cup. Remove skillet from heat and stir in remaining 2 tablespoons butter and salt and pepper to taste, stirring until butter is just incorporated. Spoon mushroom sauce around chicken and sprinkle with parsley.