Pat chicken dry; season with salt and pepper. Dust lightly with flour, shaking off excess.
Heat olive oil in a large deep skillet over medium-high heat.
Sear chicken until golden on both sides, about 6–8 minutes total; transfer to a plate.
Add peppers, onion, and mushrooms to the pan; sauté until softened and browned, 6–8 minutes.
Stir in garlic and tomato paste; cook until fragrant and slightly darkened, about 1 minute.
Pour in red wine; scrape up browned bits and reduce by half, 2–3 minutes.
Add crushed tomatoes, broth, oregano, basil, thyme, and red pepper flakes; bring to a gentle simmer.
Nestle chicken into the sauce; add capers and olives. Cover and simmer 20–25 minutes.
Uncover and cook 3–5 minutes to thicken. Taste and adjust salt. Sprinkle with parsley and serve.