Sprinkle the rabbit with salt and pepper. Heat a large Dutch oven over medium-high heat. Add the olive oil and the rabbit pieces (except for the belly), and sear on both sides until golden brown, about 4 minutes per side. Remove to a plate.
Chop the rabbit belly finely. Add to the same pot and saute the rabbit bellies with the anchovies and garlic. Use your wooden spoon to break up the anchovies, and saute until the garlic is fragrant and the anchovies are melted. Add the carrots, onions and celery and saute until tender and just beginning to turn golden. Stir in the wine and cook until the sharp wine smell cooks away, about 5 minutes. Add the tomatoes and their juices along with the olives, and give a good stir.
Add the rabbit to the pot, making sure the pieces are just covered by the tomatoes, and stir to coat with the sauce. Cover with a lid. Bring to a simmer and add the bay leaves. Simmer on low for 35 to 40 minutes. Serve sprinkled with chopped parsley.