Heat the oven to 375°F and lightly oil a 9×13-inch baking dish. Bring a large pot of well-salted water to a boil.
Cook the cavatappi 2 minutes shy of al dente. Reserve 1 1/2 cups of the pasta water, then drain and set the pasta aside.
While the water heats and pasta cooks, warm the olive oil in a large oven-safe skillet over medium-high heat. Add the mushrooms and red pepper with a pinch of salt and cook, stirring occasionally, until the edges brown and they soften, 5 to 6 minutes. Transfer to a bowl.
In the same skillet, add the ground turkey and onion. Cook, breaking up the meat, until the turkey is no longer pink and the onion is tender, 6 to 8 minutes. Stir in 1/2 teaspoon of the salt and the black pepper, then add the garlic and cook 30 seconds until fragrant.
Push the turkey to the sides to clear a hot spot in the center. Add the tomato paste and cook it, stirring, until it darkens slightly, 1 to 2 minutes. Sprinkle in the Italian seasoning and crushed red pepper. Pour in the Worcestershire and 1/2 cup reserved pasta water to deglaze, scraping up browned bits.
Add the diced tomatoes and tomato sauce. Return the mushrooms and peppers to the skillet and bring to a gentle simmer. Cook uncovered, stirring occasionally, until slightly thickened and flavorful, 6 to 8 minutes. Season with another 1/2 teaspoon salt (or to taste).
Take the pan off the heat and stir in 1 cup mozzarella and 3/4 cup Colby Jack until the sauce turns creamy. If it seems too thick, loosen with a splash of pasta water.
Fold the drained pasta into the sauce until every curve is coated. Taste and add the remaining salt if needed.
Spread the mixture in the prepared baking dish. Top with the remaining mozzarella and Colby Jack. Bake for 12 minutes, then switch to broil for 1 to 3 minutes, watching closely, until the cheese is bubbling and lightly browned.
Let rest 5 to 10 minutes so the sauce sets. Sprinkle with parsley and serve warm.