In a large pot, heat the olive oil over medium heat. Add the diced onion, garlic, bell pepper, carrots, and celery. Sauté for about 5-7 minutes until the vegetables are softened.
Add the cubed potatoes, chicken broth, Cajun seasoning, thyme, black pepper, and salt to the pot. Bring to a boil, then reduce the heat to low and let it simmer for 20-25 minutes or until the potatoes are tender.
Once the potatoes are cooked, use a potato masher or an immersion blender to mash some of the potatoes for a creamier texture, leaving some chunks for added texture.
Stir in the heavy cream and let the soup simmer for an additional 5 minutes. Adjust seasoning if necessary.
Serve hot, garnished with chopped green onions and parsley.