Pat chicken dry and season with salt and pepper. In a large heavy skillet heat oil over moderately high heat until hot but not smoking and brown chicken in 2 batches, transferring with tongs to a large plate.

Add the onions and mushrooms to the pan, until all the liquid has evaporated.  Add Marsala and cook mixture, stirring, until Marsala is almost evaporated. Add broth and chicken with any juices that have accumulated on 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 adjust seasoning to taste.  Spoon mushroom sauce around chicken and sprinkle with parsley.