Cook the pasta in a large pot of salted boiling water according to package instructions until al dente. Reserve ¼ cup pasta water before draining. Drain and set aside.
Pat the salmon filets completely dry with paper towels. Season generously on both sides with salt and freshly cracked black pepper.
In a large skillet, heat the olive oil over medium-high heat until it shimmers. Add the salmon filets skin-side up and cook for 3 to 4 minutes per side, or until golden and cooked through. Remove from the skillet and set aside.
Reduce the heat to medium. In the same skillet, add the minced garlic and cook for about 1 minute, stirring constantly, until fragrant. Do not let it brown.
Pour in the chicken broth and scrape up any browned bits from the bottom of the skillet with a wooden spoon. Let it reduce for 1 minute.
Reduce the heat to medium-low and stir in the heavy cream, capers, lemon zest, and lemon juice. Simmer for 2-3 minutes, until slightly thickened.
Flake the cooked salmon into bite-sized pieces and return to the skillet. Add the cooked pasta and toss gently until everything is well coated in the creamy lemon sauce.
Remove from heat. Sprinkle with fresh parsley and serve immediately, topped with freshly grated Parmesan cheese.