Cook rigatoni according to package directions.
While the pasta cooks, heat 1/2 tablespoons olive oil in a 12" skillet over medium-high heat. Add sausage and cook, breaking it into pieces with spatula, until it is nearly cooked. Transfer to a bowl.
Drain fat from skillet, then add remaining 2-1/2 tablespoons of oil to the skillet. Cook the shallots until they begin to soften (a minute or so).
Raise heat to medium-high and add the squash. Cook until squash is barely tender (3-5 minutes).
Reserve 1/2 cup of the water in which the pasta was cooked. Return the rigatoni to the cooking pot and add the sausage, squash mixture, and 2 tablespoons of reserve pasta water. Toss over medium heat until sausage is cooked through and rigatoni is heated through. Add more pasta water as necessary to keep dish moist.
Remove from heat, add goat cheese and parsley, and toss until the cheese melts and coats the pasta. Season to taste with salt and pepper, transfer to warm bowls, and top with Parmesan.