Cook the spaghetti or fettuccine according to package instructions in a large pot of salted boiling water until al dente. Reserve 1 cup of pasta water, then drain the pasta and set aside.
In a large skillet, heat the olive oil over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant, being careful not to burn it.
Pour in the heavy cream and bring to a gentle simmer. Stir in the grated Parmesan cheese, lemon zest, and lemon juice. Mix until the cheese is melted and the sauce is smooth.
Add the cooked pasta to the skillet and toss to coat in the creamy sauce. If the sauce is too thick, gradually add reserved pasta water until desired consistency is reached.
Season with salt and pepper to taste. Remove from heat and garnish with chopped parsley before serving.