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Pulse onion, carrot, celery, garlic, oregano, red pepper flakes, and 1/4 cup parsley in a food processor until finely chopped; transfer to a small bowl and set aside. Purée tomatoes with juices in processor; set aside.
Heat oil in a large heavy pot over medium heat; add sausage and cook, breaking up with a spoon, until browned, about 4 minutes. Add ground pork, season with salt and pepper, and cook, breaking up with a spoon, until no longer pink. Using a slotted spoon, transfer to a plate.
Increase heat to medium-high. Add reserved vegetable mixture to drippings in pot, season with salt, and cook, stirring often, until golden, 8-10 minutes.
Stir tomato paste and 1 cup water in a small bowl; add to pot. Cook, scraping up any browned bits from bottom of pot. Bring to a boil, reduce heat, and simmer until liquid has almost evaporated, 6-8 minutes.
Add reserved meat and tomato purée and 1 cup water. Bring to a boil. Reduce heat; simmer, adding more water as needed to keep meat nearly submerged, until meat is tender, about 4 hours. Season with salt.
Cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain, reserving 2 cups pasta cooking liquid.
Add pasta and 1/2 cup pasta cooking liquid to sauce; stir to coat. Stir in 3/4 cup Parmesan and remaining 1/4 cup parsley. Increase heat to medium and continue stirring, adding more pasta cooking liquid as needed, until sauce coats pasta. Divide among bowls; top with more Parmesan.
Do Ahead: Ragù can be made 3 days ahead. Let cool. Cover and chill, or freeze for up to 2 months. Reheat before using.