Bring a large pot of salted water to a rolling boil, adding 1 tablespoon salt; cook the spaghetti 1 minute less than package instructions.
Before draining, reserve 2 cups of the pasta cooking water, then drain the spaghetti in a colander.
Heat the butter in a large skillet over low heat until melted, then add the minced garlic and cook gently for about 2 minutes without browning.
Add the heavy cream to the skillet and bring it to a gentle simmer.
Sprinkle in the grated Parmesan and stir until it melts and the sauce thickens slightly, simmering about 1½ minutes.
Toss the drained spaghetti into the sauce with about 1 cup of the reserved pasta water and stir gently to coat; cook 1–2 minutes, adding more reserved water if the sauce becomes too thick.
Stir in the lemon zest, lemon juice, and chopped parsley; taste and adjust seasoning with salt and pepper as desired.
Serve immediately, optionally garnished with extra parsley and Parmesan.