In a large pot of boiling, salted water, cook 1 pound fettuccine pasta until al dente, about 8 minutes. Drain in a colander, reserving 1 cup of the pasta cooking water. Using the same pot, melt 4 tablespoons butter over medium heat. Add 1 1/2 cups heavy cream and 1/2 cup of the reserved pasta water and bring to a simmer. Add the fettuccine and 1 1/4 cups finely grated parmigiano-reggiano cheese (about 5 ounces) and toss. Season with salt and pepper and add more pasta water to make a looser sauce, if desired. Top with chopped flat-leaf parsley and serve with more cheese.