Stir the beef broth, carrots, celery, parsley, celery seed, and bay leaves into a slow cooker set on High, cover, and cook until the vegetables are tender, about 2 hours.
Place the ground beef, onion, and garlic into a skillet over medium heat. Cook, stirring frequently to break the beef into small pieces, until the onion is translucent and the beef is browned and no longer shows pink areas, 10 to 15 minutes. Drain off fat, and stir the stewed tomatoes into the beef mixture. Bring to a boil over medium heat, and cook, stirring frequently, until the tomatoes are broken up into small pieces.
Stir the potatoes, corn, green beans, and peas into the soup in the slow cooker, and add the beef mixture. Stir everything together, cover, and set the slow cooker on High. Cook for 4 hours.