Heat oil in a large skillet over medium-high heat. Add the carrots, celery, onion, and garlic; cook and stir for a few minutes, until onion is transparent. Add sausage to the skillet, and cook for a few more minutes to brown.
Pour in kidney beans, cannellini beans, and tomatoes; season with bay leaves, thyme, salt, and pepper. Cover, and reduce the heat to low. Simmer for 10 minutes, occasionally stirring until vegetables are tender.
Remove bay leaves and discard. Sprinkle parsley over the top, and serve.
