Bring a large pot of generously salted water to a boil. Cook the spaghetti until just al dente according to package directions. Before draining, reserve 1 cup of pasta cooking water. Drain and set aside.
In a large skillet over low heat, combine the olive oil, garlic, parsley, and red pepper flakes. Cook for 3 to 5 minutes, stirring occasionally, until the garlic is golden and fragrant. Do not rush this step, and do not walk away; garlic goes from golden to burned quickly.
Add the drained spaghetti to the skillet and toss to coat evenly in the garlic oil. Stir in the melted butter until everything is well combined. Add a splash of reserved pasta water if needed to loosen the sauce.
Season with salt and black pepper, top with freshly grated Parmesan, and serve immediately.