In a medium saucepan, cover the potatoes with cold water. Bring to a boil, then turn down heat and cook until potatoes are fork tender, about 10 minutes. Drain and cool slightly, then coarsely chop.

In a 10- or 12-inch skillet, melt the butter, add the onions, and cook for 3 to 5 minutes. Add the olive oil and the chopped potatoes and cook over medium heat until the potatoes are lightly browned, about 8 minutes. Add the apple, chilies, and cream. Continue to cook for 5 minutes and remove from heat. Break the salmon into 2-inch pieces and place on top of the potatoes. Sprinkle with parsley, salt, and pepper; top with sour cream and green onions; and serve.