Melt 1 tablespoon butter in a 12-inch sauté pan over medium-high heat. Add shrimp, season with salt and pepper, and cook, stirring occasionally, until opaque and cooked through, about 2–3 minutes per side depending on size; transfer to foil and keep warm.
Reduce heat to medium and melt the remaining 2 tablespoons butter in the same pan. Add the minced garlic and sauté about 20 seconds until fragrant and slightly golden.
Add 3 cups water, 1 cup half-and-half, and the linguine to the pan. Season with salt and pepper to taste and bring to a boil over medium-high heat.
Boil, stirring frequently, until the noodles reach desired doneness, about 10 minutes; there should be some liquid remaining, which will become the sauce.
Remove the pan from the heat, add the shredded Parmesan, and stir until the cheese melts into a creamy sauce; it may clump briefly but will smooth with stirring.
If the sauce is too thick, thin with additional half-and-half a little at a time until desired consistency is reached. Return the cooked shrimp to the pan and toss to combine.
Sprinkle chopped parsley over the pasta and serve warm.