Melt the butter in a large skillet over medium heat. Add the minced garlic and sauté, stirring, until fragrant, about 1 minute—do not let the garlic brown.
Add the crab meat, heavy whipping cream, Parmesan cheese, salt, and black pepper to the skillet. Reduce the heat to low and stir constantly until the cheese melts and the sauce is smooth and heated through, about 2–3 minutes.
Add the hot, cooked and drained fettuccine to the sauce. Toss gently in the skillet until the noodles are evenly coated.
Remove from the heat, divide among plates, garnish with the chopped fresh Italian parsley, and serve immediately.