In a medium-sized saucepan, boil water and salt it to your preference. Add your pasta, and cook according to the package instructions until al dente. When the pasta is cooked, reserve 1 cup of the pasta water and drain the pasta.
Meanwhile, pat the salmon dry and season with salt and pepper.
In a skillet over medium heat, melt the butter. Cook the salmon on all sides until it has a nice crust and is cooked through. The salmon is done when the internal temperature reaches 145°F/62°C. Remove onto a plate and keep warm.
To the same skillet, add the garlic and saute for 30 seconds. Then deglaze the pan with reserved pasta water (about ¼ a cup) and scrape off any bits stuck to the bottom of the pan.
Add the heavy cream, lemon juice, and parmesan. Let the sauce bubble until it becomes smooth. Add additional reserved pasta water if needed to get the sauce to the right consistency. Taste it and adjust seasonings as needed.
Add the salmon back to the sauce, and gently break it apart with your spatula into bite sized pieces.
Add the cooked pasta, and mix it in with the sauce and the salmon. Cook for one more minute until everything is warmed through.
Remove from heat, garnish with fresh parsley and serve immediately with extra grated parmesan cheese and a squeeze of fresh lemon.