Bring a large pot of well-salted water to a boil. Cook the pasta until just al dente according to the package directions. Reserve 1/2 cup pasta water, then drain.
Pat the shrimp dry and season with the salt and black pepper. Heat 1 tablespoon olive oil and 2 tablespoons butter in a large skillet over medium-high heat.
Add the shrimp in a single layer and cook for 1 to 2 minutes per side, until pink and opaque. Transfer to a plate; do not overcook.
Lower the heat to medium. Add the remaining olive oil and butter, then cook the garlic and red pepper flakes for about 30 seconds, just until fragrant.
Pour in the cream and bring it to a gentle simmer. Stir in the Parmesan until smooth, loosening the sauce with reserved pasta water a splash at a time.
Add the drained pasta and toss until coated. Return the shrimp to the skillet and warm for about 1 minute.
Taste and adjust seasoning. Finish with parsley and extra Parmesan, then serve immediately.
Notes
The sauce should coat the pasta while remaining loose enough to toss; add reserved pasta water as needed.