Whisk flour, baking powder, fine salt, and garlic powder in a medium bowl.
Fold in cheddar and 1 tablespoon parsley. Stir in milk and melted butter until just combined. Let rest 10 minutes.
Warm olive oil in a large pot over medium heat. Soften onion, carrot, and celery with a pinch of salt, 6–8 minutes.
Stir in garlic and tomato paste. Cook until the paste darkens and smells sweet, 1–2 minutes.
Pour in crushed tomatoes and broth. Add Italian seasoning, sugar, red pepper flakes, kosher salt, and black pepper. Bring to a gentle simmer.
Reduce heat to maintain a low simmer. Drop heaping tablespoons of dough into the soup, spacing them out.
Cover and cook without peeking until dumplings are puffed and cooked through, 12–15 minutes.
Stir in heavy cream and Parmesan. Simmer 2 minutes more and adjust seasoning to taste.
Ladle into bowls and sprinkle with remaining parsley. Add extra Parmesan if you like.