Boil a large pot of salted water. Cook pasta until al dente. Reserve 1/2 cup cooking water and drain.
Season salmon with 1/4 tsp salt and pepper. Heat olive oil in a skillet over medium-high and sear 3–4 minutes per side.
Transfer salmon to a plate and flake into large chunks. Reduce heat to medium.
Melt butter in the same pan. Sauté garlic until fragrant, about 30 seconds.
Pour in heavy cream and a splash of the pasta water. Simmer gently until slightly thickened, 2–3 minutes.
Stir in lemon zest, lemon juice, and red pepper flakes. Whisk in Parmesan until smooth and glossy.
Add spinach to the sauce and wilt. Toss in drained pasta and most of the parsley, loosening with more pasta water as needed.
Gently fold in flaked salmon. Taste and adjust salt and pepper. Serve hot, topped with remaining parsley.