In a large, deep skillet, heat 2 tablespoons of the olive oil. Add the squash, season lightly with salt and cook over high heat, stirring, until tender and lightly browned, about 8 minutes. Using a slotted spoon, transfer the squash to a plate. Add the remaining 1 tablespoon of olive oil to the skillet. Add the onion and sausage and cook over high heat, breaking up the sausage with a spoon, until lightly browned and cooked through. Return the squash to the skillet.
Meanwhile, in a large pot of boiling salted water, cook the pasta until al dente. Drain the pasta, reserving 1 cup of the cooking water. Add the pasta to the skillet along with the reserved cooking water and the cheese and cook over moderately high heat, stirring, until the sauce is thick and creamy, 1 to 2 minutes. Transfer the pasta to a bowl, sprinkle with the parsley and serve right away.