Bring a large pot of salted water to a boil and cook the spaghetti according to package directions until al dente; reserve 1 cup of the pasta cooking water, then drain the pasta.
While the pasta cooks, warm a large deep skillet over low heat and add the olive oil, chopped garlic, and crushed red pepper flakes; cook for about 1 minute, stirring, until fragrant but not browned.
Add 3/4 cup of the reserved pasta water, the lemon juice, and the heavy cream to the skillet; increase heat to medium and bring to a gentle boil, stirring frequently, until the sauce thickens slightly, about 2–3 minutes.
Stir in the lemon zest and half of the grated Parmesan; season with the kosher salt and black pepper, tasting and adjusting as needed.
Add the drained spaghetti to the skillet, remove from heat, and toss with tongs for 1–2 minutes so the pasta absorbs the sauce; add a splash of reserved pasta water if needed to loosen the sauce.
Serve topped with the remaining Parmesan and the chopped parsley.