Bring a large pot of salted water to a rolling 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, heat a large deep skillet over low heat and add the olive oil, chopped garlic, and crushed red pepper; 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, 2–3 minutes.
Stir in the lemon zest and half of the grated Parmesan; season with the kosher salt and black pepper to taste.
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 the sauce seems too thick.
Transfer to plates or a serving bowl, sprinkle with the remaining Parmesan and chopped parsley, and serve immediately.
