Stir tomato sauce, broth, tomato paste, Italian seasoning, brown sugar, and red pepper flakes together; set aside.
Combine beef, rice, beaten egg, milk, onion, garlic, Worcestershire, salt, pepper, and parsley in a bowl. Mix gently.
Shape the mixture into about 16 golf-ball-size meatballs with slightly damp hands.
Warm the olive oil in a large deep skillet or Dutch oven over medium heat.
Brown meatballs in batches, turning to color several sides, 4–5 minutes total. Do not cook through.
Pour in the prepared tomato mixture. Scrape up browned bits and bring to a gentle simmer.
Nestle meatballs into the sauce, cover, and simmer on low until rice is tender, 30–35 minutes. Turn once halfway.
Rest 5 minutes. Sprinkle with extra parsley and serve with mashed potatoes, pasta, or crusty bread.