Bring a large pot of water to a boil. Cook the pasta according to package directions. Drain the pasta, reserving 1 cup of the pasta cooking liquid.
Heat the oil in a large deep nonstick skillet over medium-high heat. Add the mushrooms, turkey, tomatoes, vinegar, onion, crushed red pepper and 3/4 teaspoon salt and cook, stirring and breaking up the meat with a wooden spoon, until the vegetables are soft and the turkey is cooked through, about 8 minutes. Add 1/2 cup of the reserved pasta water, bring to a simmer and cook until the consistency of tomato sauce, about 4 minutes.
Add the cooked pasta to the turkey-mushroom sauce along with the parsley. Toss to combine; adjust seasoning if needed.
Transfer the pasta to a serving platter or individual bowls. Sprinkle with the cheese and parsley and serve.
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