Bring a large pot of salted water to a boil and cook the fettuccine until al dente according to package instructions. Before draining, reserve about 1 cup of the pasta water, then drain the pasta and set aside.
While the pasta cooks, pat the shrimp dry with paper towels and season lightly with salt and black pepper.
Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Once hot, add the shrimp in a single layer and cook for 1 to 2 minutes per side, or until lightly golden, pink, and just cooked through. Transfer the shrimp to a plate and set aside.
In the same skillet, melt the remaining 4 tablespoons butter. Add the garlic and cook for about 30 seconds, just until fragrant.
Sprinkle in the flour and whisk continuously for 1 minute to form a smooth roux.
Slowly pour in the heavy cream, whisking until smooth. Bring to a gentle simmer and cook for 2 to 3 minutes, or until slightly thickened (do not boil).
Reduce the heat to low and stir in the Parmesan cheese until melted and smooth.
Add the cooked fettuccine to the sauce and toss until evenly coated. Add a splash of the reserved pasta water as needed to loosen the sauce.
Return the shrimp to the skillet and gently toss to combine. Garnish with chopped parsley and additional black pepper, if desired, then serve right away.