Bring 6 cups water to a boil in a large saucepan.
Using a small paring knife, remove the vein along the back of each shrimp and rinse under running water. Add the garlic to the boiling water and cook for 2 minutes. Add the shrimp to the water with the garlic and cook for 2 to 3 minutes. Drain the shrimp and garlic. Peel and finely chop the garlic.
Heat the oil in a large skillet over medium heat. Add the garlic and cook for 1 minute. Add the heavy cream and bring to a simmer. Cook, stirring occasionally, until the cream begins to thicken. Stir in the Parmesan, parsley and shrimp. Season with salt and pepper.
Pour the sauce over the linguine in a large bowl and toss to coat. Garnish with more Parmesan and parsley.