Heat the olive oil in a skillet over medium heat. Add the sausage slices and cook until lightly browned on both sides.
Add the onion, garlic, and carrots to the skillet. Cook for 3 to 4 minutes until the vegetables begin to soften.
Transfer the sausage and vegetable mixture to the slow cooker.
Add the cannellini beans, diced tomatoes with their juices, chicken broth, thyme, bay leaf, salt, and black pepper. Stir everything together.
Cover and cook on low for 6 to 7 hours until the beans are soft and the flavors have blended.
Remove the bay leaf and stir in the fresh parsley just before serving.