Mix flour and parsley in a bowl; season with salt and pepper. Dip veal shanks into flour mixture, shaking off excess flour.
Heat olive oil in a large skillet over medium heat. Fry veal in hot oil until browned, about 10 minutes; transfer to a large baking dish. Return skillet to stove.
Cook and stir onion, carrots, and celery in the skillet used to fry veal until vegetables begin to brown, about 5 minutes. Transfer vegetables to the baking dish and pour wine into the skillet. Bring to a boil, scraping up any browned bits from the skillet. Reduce heat to medium-low and continue to simmer until wine is reduced by half, about 5 minutes.
Stir tomatoes, thyme, and bay leaves into wine. Simmer until tomatoes are hot, about 10 minutes. Pour tomato mixture over veal and vegetables. Stir to combine. Cover baking dish with aluminum foil.
Bake in the preheated oven until veal is very tender, about 90 minutes.