Season rabbit all over with half the salt and pepper. Dust lightly with flour and shake off excess.
Heat olive oil in a large Dutch oven over medium-high heat. Brown rabbit in batches until golden; set aside.
Lower heat to medium. Sauté onion, bell pepper, and mushrooms in the pot until softened, about 6–8 minutes.
Stir in garlic and tomato paste. Cook 30 seconds, just until fragrant.
Pour in white wine and scrape up browned bits. Simmer until reduced by roughly half, 2–3 minutes.
Add crushed tomatoes, broth, oregano, thyme, rosemary, red pepper flakes, and sugar. Stir to combine.
Nestle rabbit and any juices into the sauce. Cover and simmer gently until tender, 35–40 minutes.
Stir in olives and capers. Uncover and simmer 5–8 minutes to slightly thicken the sauce.
Finish with red wine vinegar and parsley. Taste and adjust salt and pepper before serving.