Heat 1 tablespoon olive oil in a large wide pot over medium-high heat. Add the sausage and cook, breaking up the meat into small pieces with a wooden spoon, until lightly browned and no longer pink, about 5 minutes. Pour off the fat from the meat.
Add the remaining 1 tablespoon olive oil and the onion to the skillet; cook, stirring occasionally, until the onion softens, about 5 minutes. Stir in the carrot, garlic and tomato paste and cook until the garlic softens, 1 to 2 minutes.
Add 4 1/2 cups water, the spaghetti, cherry tomatoes, 1/2 teaspoon salt and a few grinds of pepper to the pot. Increase the heat to high and bring to a boil, then reduce the heat to a simmer; cook, stirring occasionally, until the pasta is tender and the sauce thickens, 10 to 12 minutes. Remove from the heat and add the parsley; season with salt and pepper and toss.