Bring a large pot of salted water to a boil. Cook fettuccine according to package directions until al dente. Before draining, reserve 1 cup of pasta water. Drain pasta and set aside.
In a large skillet over medium-low heat, melt the butter. Add minced garlic and sauté for 30 seconds until fragrant, but not browned.
Pour in the heavy cream, stirring constantly. Gradually add the grated Parmesan, whisking until melted and smooth. Season with salt and pepper.
Add the drained fettuccine to the skillet. Toss with tongs, adding a splash of reserved pasta water as needed to thin the sauce and help it cling to the noodles.
Remove from heat, sprinkle with parsley, and toss once more. Serve immediately in warmed bowls with extra Parmesan on top.