Stir beef broth, carrots, celery, parsley, celery seed, and bay leaves into a slow cooker set on High, cover, and cook until vegetables are tender, about 2 hours.

Place ground beef, onion, and garlic into a skillet over medium heat. Cook, stirring frequently to break the beef into small pieces, until onion is translucent and beef is browned and no longer shows pink areas, 10 to 15 minutes. Drain off fat, and stir 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 potatoes, corn, green beans, and peas into soup in slow cooker, and add beef mixture. Stir everything together, cover, and set slow cooker on High. Cook for 4 hours.