In a heavy 12-inch skillet melt 2 Tbsp. of the butter over moderate heat. Add the onions and saute 5 to 7 minutes, stirring often, until limp and golden; transfer to a large bowl and reserve.
Melt 2 more Tbsp. of the butter in the skillet, add the apples, and saute about 5 minutes, stirring often, until golden; transfer to a separate bowl and reserve.
Dredge the liver in the flour, shaking off any excess. Melt the remaining butter in the skillet over high heat and, as soon as it foams up and subsides, add half the liver and brown 2 to 3 minutes on a side, depending on whether you like liver rare or medium-rare. (I like mine well done, so I will cook longer, covered.)
Lift the browned slices to a heated platter and sprinkle each lightly with salt and pepper. Tent with foil to keep warm. Add the remaining liver to the skillet and brown as before. Transfer to the platter and season with salt and pepper.
Pile the onions and apples on top of the liver, sprinkle with the parsley and serve.