Start by bringing a large pot of salted water to a boil. Add the spaghetti and cook according to package instructions until al dente. Reserve about 1/2 cup of the pasta water, then drain the spaghetti and set aside.
While the spaghetti cooks, season the salmon fillets with cracked black pepper and a pinch of salt. In a skillet, heat the olive oil and butter over medium heat. Once hot, add the salmon fillets and cook for about 4-5 minutes on each side or until the salmon is cooked through and flakes easily with a fork.
Remove the salmon from the skillet and set aside. In the same skillet, add the heavy cream, juice of one lemon, and the zest of both lemons. Stir well, allowing the mixture to simmer for a few minutes until it thickens slightly.
Flake the cooked salmon into bite-sized pieces and add it back into the skillet with the creamy sauce. Toss in the cooked spaghetti, adding reserved pasta water a little at a time until you reach your desired sauce consistency. Stir in the chopped parsley and adjust seasoning with additional salt and pepper if necessary.
Serve the Creamy Lemon Salmon Pasta warm, garnished with extra parsley or lemon zest if desired. Enjoy this delightful dish with a glass of white wine or a refreshing sparkling water!