Cut the pasta dough into tagliatelle.
In a large pot of boiling, salted water, cook the tagliatelle, stirring often, until al dente, about 3 minutes.
Meanwhile, in a large, heavy skillet over medium-high heat, melt the butter. Stir in 1 cup of the pasta cooking water. Using a spider or small strainer, transfer the pasta from the pot of water to the skillet. Toss with half of the cheese, adding more pasta water as needed to form a creamy sauce. Toss with the parsley and half of the walnuts; season with salt and pepper. Sprinkle with the remaining cheese and walnuts.