Heat the olive oil in a 4-quart saucepan over medium-high heat. Add the onion, garlic, thyme, carrots, celery, and 1/2 teaspoon salt. Cook, stirring occasionally, until softened, 5 to 7 minutes. Add the tomato paste and cook 1 minute, stirring to coat the vegetables. Add the beef, 1 teaspoon salt, and 1/4 teaspoon pepper. Cook, breaking up the meat with a spoon, until browned, about 5 minutes.
Add the canned tomatoes, bucatini, 3 cups water and 1 teaspoon salt. Stir to ensure the pasta is submerged in liquid. Bring to a boil and continue to cook, stirring frequently to keep the pasta from sticking to the bottom of the pot, until the pasta is al dente and the sauce has thickened, 10 to 12 minutes. Remove the thyme sprigs. Stir in the Parmesan and parsley and season with salt to taste. Divide among 4 bowls and serve with additional parsley and Parmesan.
Cook's Note: The success of this recipe is dependent upon the specific pasta shape suggested, so we don't recommend substitutions.
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