Bring a large pot of water to a boil and add 1 tablespoon salt. Add the spaghetti and cook until al dente, about 1 minute less than the package time.
Before draining, scoop out 1 cup of the starchy pasta water and set it aside. Drain the pasta.
While the pasta cooks, melt the butter in a large deep skillet over medium-low heat. Add the garlic and red pepper flakes, if using, and cook, stirring often, for 1 to 2 minutes, until fragrant and pale golden. Do not let the garlic brown.
Pour in the cream and bring it to a gentle simmer. Simmer for 2 to 3 minutes, stirring occasionally, until slightly thickened.
Turn the heat to low. Add the drained pasta and 1/2 cup of the reserved pasta water, and toss with tongs to coat.
Sprinkle in the Parmesan a handful at a time, tossing continuously until it melts into a smooth, glossy sauce. Add more pasta water, a splash at a time, if the sauce looks thick.
Season with the 1/4 teaspoon salt and the black pepper, then taste and adjust. Toss in most of the parsley.
Serve right away, topped with the remaining parsley and extra Parmesan.