Bring a large pot of salted water to a boil and cook spaghetti until al dente. Reserve about 1 cup of pasta water before draining. This starchy water helps the sauce cling beautifully to the noodles.
Melt butter in a large skillet over medium heat. Add grated garlic and red pepper flakes, stirring until fragrant, about 1–2 minutes. Be careful not to burn the garlic, as it will taste bitter.
Pour in the heavy cream, then add lemon zest, lemon juice, thyme, and parsley. Bring the mixture to a gentle boil and then reduce to a simmer for 10 minutes, stirring occasionally. The sauce should thicken slightly and coat the back of a spoon.
Stir in freshly grated Parmesan cheese and season with salt and pepper. The cheese will melt into the sauce, making it rich and creamy. Adjust seasoning to taste.
Add the cooked pasta directly into the sauce and toss gently. Use reserved pasta water a tablespoon at a time to achieve your desired sauce consistency. Ensure every strand is coated in the creamy, zesty sauce.
Turn off the heat and sprinkle extra chopped parsley and Parmesan over the pasta. Serve immediately for the best flavor and creamy texture.