Heat oil in a large nonstick skillet over medium heat. Add onion, carrot, parsnip, and garlic; cover and cook 5 minutes or until tender.
Place in a 5-quart electric slow cooker. Add beans and next 6 ingredients (through bay leaf). Cover and cook on low 8 hours or until vegetables are tender.
Combine breadcrumbs, cheese, and butter in a small bowl; toss with a fork until moist. Stir breadcrumb mixture and sausage into bean mixture; sprinkle with parsley.