Bring a large pot of water to a boil for the pasta.
In a large skillet, crumble the sausage. Add the carrots and saffron and cook over mediumhigh, breaking up the sausage with a wooden spoon, until the sausage is browned and cooked through, about 6 minutes.
Salt the water, add the pasta and cook until al dente. Drain the pasta, reserving 1 ⁄2 cup starchy cooking water. Stir the pasta and reserved cooking water into the sausage mixture, then add the lemon juice and butter, tossing to coat. Season with salt and pepper.
Divide among 4 shallow bowls; top with the parsley and cheese.