Start by bringing a large pot of salted water to a boil. Add the linguine pasta and cook according to package instructions until al dente. Reserve 1/2 cup of the pasta cooking water, then drain the pasta in a colander and set aside.
While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the minced garlic and red pepper flakes. Sauté for about 1 minute until fragrant, being careful not to burn the garlic.
Add the shrimp to the skillet and season with salt and pepper. Cook for about 3-4 minutes, stirring occasionally, until the shrimp turns pink and is cooked through.
Once the shrimp is cooked, stir in the lemon zest, lemon juice, and broth. Bring the mixture to a gentle simmer and let it cook for an additional 2-3 minutes to meld the flavors.
Add the drained linguine to the skillet, along with the reserved pasta water as needed to loosen the sauce. Toss everything together to combine well.
Remove from heat and garnish with fresh parsley. Serve hot with grated Parmesan cheese if desired.