Authentic Alfredo Sauce with Fettuccine

Bahan-bahan (4 porsi)

• ½ cup (1 stick) unsalted butter, (cut into pieces)
• ¾ cup freshly grated Parmesan cheese, (divided)
• ¾ cup starchy pasta water
• 16 ounces fettuccine pasta ((dry or fresh))
• kosher salt and freshly ground black pepper, (to taste)
• additional Parmesan and chopped fresh Italian parsley, (for garnish)