In a large pot, heat oil (2 tablespoons) over medium-high heat. Stir in onions (1), garlic (3 cloves), thyme (3 sprigs), carrots (2), celery (2 stalks), and 1/2 teaspoon salt, and saute until onions are translucent. Stir in tomato paste (2 tablespoons), stirring to coat the vegetables, and cook for 1 minute.
Add the sausage (1 pound), 1/2 teaspoon salt, and 1/4 teaspoon pepper. Brown the sausage, breaking it up into large chunks with a spoon, until browned, about 5 minutes.
Pour the canned tomatoes (1 can) and 3 cups of water over the sausage mixture. Add bucatini (12 ounces) and salt stirring to ensure the bucatini is submerged.
Bring to a boil, stirring often, until the bucatini is cooked to al dente, about 10-12 minutes. Add parmesan (1/4 cup) and adjust the seasoning. Garnish with parsley (1/4 cup) and serve.