Heat olive oil in a large pot or Dutch oven over medium heat. Add diced onion, celery, and carrots, and saut� for about 5 minutes until softened.
Add minced garlic to the pot and cook for an additional 1 minute until fragrant.
Pour in the chicken or vegetable broth and diced tomatoes with their juices. Stir in the diced potatoes, frozen corn, frozen lima beans, bay leaves, and Old Bay seasoning.
Bring the soup to a boil, then reduce the heat to low and let it simmer uncovered for about 20-25 minutes until the vegetables are tender.
Once the vegetables are cooked, stir in the crab meat and simmer for an additional 5 minutes until heated through. Season with salt and pepper to taste.
Remove the bay leaves from the soup and discard.
Ladle the hot Maryland Crab Soup into bowls, garnish with chopped fresh parsley if desired, and serve hot.