Preheat oven to 500 degrees F.
Cut potato into 1/4-inch dice. In a saucepan of boiling salted water simmer potato 5 minutes, or until barely tender, and drain in a colander. In a heavy skillet cook bacon and onion in 1/2 tablespoon oil over moderate heat, stirring occasionally, until onion is softened, about 5 minutes. Add potato and cook, stirring occasionally, 5 minutes, or until golden tender. Cool mixture slightly and stir in turkey cheese, and chopped parsley.
Put 2 tortillas on a baking sheet and brush with some remaining oil. Turn tortillas over and top each with half of turkey hash, spreading evenly. Cover with remaining 2 tortillas and brush tops of quesadillas with remaining oil.
Bake quesadillas in middle of oven until golden, 8 to 10 minutes. Transfer quesadillas to a cutting board and cut into wedges. Top wedges with cranberry sauce and parsley leaves.