Bring a large pot of salted water to a boil. Add the linguine and cook according to package instructions until al dente, usually about 8-10 minutes. Reserve 1/2 cup of pasta water before draining the linguine in a colander. Set aside.
Heat olive oil in a large skillet over medium-high heat. Add the peeled and deveined shrimp in a single layer. Season lightly with salt and black pepper. Cook for 2-3 minutes per side or until the shrimp turn pink and opaque. Remove shrimp from the skillet and set aside.
In the same skillet, reduce heat to medium. Add the minced garlic and sauté for about 1 minute until fragrant but not browned. Pour in the heavy cream and stir to combine. Let the cream simmer gently for 2-3 minutes to thicken slightly.
Stir in the grated Parmesan cheese until melted and the sauce becomes smooth. Add the lemon zest and lemon juice, mixing well. Taste and adjust seasoning with salt and pepper as needed.
Return the cooked shrimp to the skillet. Add the drained linguine and toss everything together. If the sauce seems too thick, pour in some reserved pasta water a little at a time until you reach your desired consistency.
Remove from heat and sprinkle chopped fresh parsley over the top. Serve immediately with extra Parmesan on the side, if desired.
