Bring a large pot of salted water to a rolling boil and cook the spaghetti until al dente according to package directions; reserve 1 cup of the pasta cooking water, then drain the pasta.
While the pasta cooks, heat a large deep skillet over low heat. Add the olive oil, chopped garlic, and crushed red pepper flakes and cook, stirring, for about 1 minute 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, then season the sauce with kosher salt and freshly ground black pepper to taste.
Add the drained spaghetti to the skillet, remove from the 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.
Transfer to serving bowls and top with the remaining Parmesan and chopped parsley, then serve immediately.