Cook the pasta al dente in salted water according to the instructions on the package. Reserve 1-2 cups of pasta water before draining.
Meanwhile, heat the olive oil in a large pan or Dutch oven and melt the butter in it over low-medium heat.
Stir in the garlic and red chili flakes and cook for 1-2 minutes until the garlic changes its color, careful not to burn it.
Transfer the cooked pasta to the pan and toss well with a pair of kitchen tongs to cover it in the butter sauce. Add a splash of pasta water if it looks too thick.
Stir in the lemon zest and juice, then add the Parmesan and toss well to combine.
Add the chopped parsley, season to taste, and serve immediately, topped with additional grated cheese and lemon zest if you like.