Bring a pot of salted water to boil. Add the spaghetti and cook according to package instructions. Reserve 1 1/2 cups cooking liquid, then drain.
Meanwhile, heat 1 tsp. oil in a large nonstick skillet over medium heat. Add the sausage and cook, breaking it up with a wooden spoon, until browned , about 5 minutes. Transfer to a bowl.
Add the remaining 2 tsp. oil to the skillet and increase the heat to medium-high. Add the mushrooms, season with salt and pepper and cook until lightly browned, 3-4 minutes.
Stir the tomato paste and rosemary into mushrooms and cook, stirring 1 minute. Return the sausage to the skillet, then add the cream and 1 cup reserved pasta water; bring to a gentle simmer and cook until slightly thickened, 8-10 minutes. Season with salt and pepper.
Add the spaghetti to the skillet and toss with the sauce mixture, adding some of the reserved 1/2 cup cooking water to loosen, if needed. Top with shredded parmesan.