Bring a large pot of salted water to a boil and cook the linguine until al dente according to the package instructions. Reserve ½ cup of pasta water before draining.
While the pasta cooks, season the shrimp with salt and black pepper.
Heat the butter and olive oil in a large skillet over medium heat.
Add the shrimp and cook for 1–2 minutes per side until pink and opaque. Transfer to a plate.
In the same skillet, add the minced garlic and cook for about 30 seconds until fragrant.
Stir in the heavy cream, Italian seasoning, lemon juice, and lemon zest. Simmer gently for 2–3 minutes.
Add the Parmesan cheese and stir until the sauce becomes smooth and creamy.
Return the shrimp to the skillet and toss to coat in the sauce.
Add the cooked linguine and gently toss everything together. If the sauce is too thick, stir in a little reserved pasta water until it reaches your desired consistency.
Sprinkle with fresh parsley and serve immediately with extra Parmesan and additional lemon wedges if desired.