In a plastic bag combine 1/2 cup flour along with the salt and pepper.
Rinse the rabbit pieces and pat them dry with paper toweling. Shake the rabbit pieces in the plastic bag to coat-shaking off any excess.
Place 3 tablespoons of the oil in a Dutch oven or chicken cooker and brown the rabbit on both sides. Set aside.
In the same pan add the other tablespoon of olive oil if needed and cook the onions until they start to brown. Mix in the garlic, thyme, basil, and rosemary, and continue cooking for about 1 minute.
Mix in the beer and chicken broth while stirring to scrape any dripping from the bottom of the pan.
Add the rabbit back into the pan, bring to a boil, and then reduce heat to a medium simmer. Cook covered until the rabbit is tender; about 30 minutes. (It may take a little longer for wild rabbit)
Remove the rabbit to a serving platter. Mix together the 3 tablespoons of flour and melted butter to form a thickening for the gravy, skim off any excess fat if needed, and then stir the mixture into the juices; cook until thickened.
Salt and pepper to taste, stir in the parsley, and serve the gravy over the rabbit.