Brown the beef in batches. Heat a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the ground beef in a single layer, breaking it into large chunks with a wooden spoon. Let it sit undisturbed for 3–4 minutes to develop a dark crust before stirring. Cook until well browned, about 8–10 minutes total. If your pot is not large enough to hold all the beef in one layer, brown it in two batches. Transfer the browned beef to a plate.

Drain the fat. Carefully tilt the pot and spoon off the excess fat, leaving about 1 tablespoon in the pot. Discard the drained fat once it cools. Do not wipe the pot — the browned bits on the bottom are flavor.

Soften the aromatics. Return the pot to medium heat. Add the olive oil if the pot is dry. Add the diced onion, celery, and carrots. Cook, stirring occasionally, until the onions are translucent and the carrots begin to soften, about 5–6 minutes. Scrape up any browned bits from the bottom of the pot as the vegetables release moisture.

Bloom the garlic and tomato paste. Add the minced garlic and cook for 30 seconds until fragrant. Add the tomato paste and cook, stirring constantly, for 2 minutes until it darkens to a deep brick red. This step caramelizes the tomato paste and builds a richer broth.

Build the broth. Pour in the beef broth, scraping the bottom of the pot to dissolve any remaining fond. Add the diced tomatoes with their juices, Worcestershire sauce, oregano, thyme, smoked paprika, salt, pepper, cayenne, and the Parmesan rind if using. Stir to combine.

Return the beef and add potatoes. Add the browned ground beef and any accumulated juices back to the pot. Add the red potatoes. Stir to combine. Bring the soup to a boil, then reduce the heat to a gentle simmer. Cover and cook for 15 minutes.

Add the remaining vegetables. Stir in the diced bell peppers, frozen corn, and frozen green beans. Cover and simmer for 10–12 minutes more, until the potatoes are fork-tender and the peppers are softened but still hold their shape.

Finish with vinegar and herbs. Remove the Parmesan rind. Stir in the red wine vinegar. Taste and adjust salt and pepper. The vinegar should brighten the broth without making it taste sour. Start with 1 tablespoon and add more if needed.

Rest and serve. Let the soup rest off the heat for 5 minutes. This lets the flavors settle and the broth thicken slightly. Ladle into bowls and top with fresh chopped parsley. Serve with crusty bread, cornbread, or oyster crackers.