Bring a large pot of water to a boil. Add the 12 oz spaghetti and the 1/4 teaspoon salt; cook until al dente according to package directions. When done, reserve 1 cup of the pasta cooking water, then drain the pasta and set it aside.
While the water is heating (or while the pasta cooks), cut the 2 salmon fillets into bite-sized pieces. Season the salmon evenly with the 1/2 teaspoon cracked black pepper.
Zest one of the 2 lemons and set the zest aside. Cut both lemons in half and juice them. Reserve the juice of 1/2 lemon to squeeze over the cooked salmon; set aside the remaining 1 1/2 lemons' juice for the sauce.
Heat a large nonstick skillet over medium-high heat. Add the 1 teaspoon butter and 1 tablespoon olive oil; heat until the butter has melted and begins to foam.
Add the salmon pieces in a single layer (work in batches if they crowd the pan). Cook undisturbed about 1.5–2 minutes, flip, and cook another 1–2 minutes until a light crust forms and the salmon is almost cooked through.
Transfer the salmon to a warmed plate, squeeze the reserved 1/2 lemon juice over it, tent with foil, and set aside.
Pour the 1 cup heavy cream into the same skillet. Add the reserved lemon zest and the juice of the remaining 1 1/2 lemons. Bring to a gentle simmer over medium heat and cook, stirring occasionally, about 2 minutes until the sauce thickens slightly.
Stir in 1/4 cup of the reserved pasta cooking water to loosen the sauce. Return the drained pasta to the skillet and toss gently to coat; add more reserved pasta water, a tablespoon at a time, if the sauce needs to be thinner.
Gently fold the cooked salmon into the pasta and sauce to warm through, taking care not to break up the pieces.
Divide the pasta among plates and sprinkle with the 3 tablespoons coarsely chopped parsley. Serve immediately.