In a mixing bowl, combine the rabbit pieces, 2 tablespoons of olive oil, minced garlic, dried thyme, and a pinch of salt and pepper. Toss well to coat the meat evenly. Let it marinate for at least 30 minutes, or refrigerate overnight for more flavor.
Heat the remaining 2 tablespoons of olive oil in a large, heavy-bottomed pot or Dutch oven over medium-high heat. Brown the marinated rabbit pieces on all sides until they develop a golden-brown crust. Work in batches if needed to avoid overcrowding the pot. Remove the browned rabbit pieces and set them aside.
In the same pot, add the diced onion, carrots, and celery. Cook over medium heat until the vegetables soften and start to brown slightly, about 5 minutes.
Return the browned rabbit pieces to the pot and add the bay leaf. Pour in the chicken or vegetable broth and red wine, if using. The liquid should cover the rabbit pieces, but not completely submerge them. Bring the liquid to a simmer.
Return the browned rabbit pieces to the pot and add the bay leaf. Pour in the chicken or vegetable broth and red wine, if using. The liquid should cover the rabbit pieces, but not completely submerge them. Bring the liquid to a simmer.
Once the rabbit is cooked, remove the pot from the heat. Taste the stew and adjust the seasoning with salt and pepper as needed.
Serve the rabbit stew hot in bowls or plates. Garnish with chopped fresh parsley, if desired. Accompany the stew with crusty bread or mashed potatoes to soak up the flavorful broth.