Cook the pasta according to package instructions until al dente. Drain and set aside, reserving 1/2 cup of pasta water.
In a large skillet, heat the olive oil over medium heat. Season the salmon fillet with 1/2 teaspoon salt and black pepper. Add the salmon to the skillet and cook for about 4-5 minutes on each side, or until cooked through. Remove from the skillet and flake into bite-sized pieces.
In the same skillet, add the minced garlic and sauté for about 1 minute until fragrant. Add the fresh spinach and cook until wilted, about 2-3 minutes.
Pour in the heavy cream and bring to a gentle simmer. Stir in the Parmesan cheese, lemon juice, and lemon zest. Mix until the cheese is melted and the sauce is creamy.
Add the cooked pasta and flaked salmon to the skillet. Toss to combine, adding reserved pasta water as needed to reach desired sauce consistency. Adjust seasoning with remaining salt if necessary.
Serve immediately, garnished with fresh parsley.