In a large pot over medium heat, melt butter and saute carrots, celery and onion until softened, about 6 minutes. Add the flour to the vegetables and blend well. Cook the flour until it achieves a slight blonde color. Next, stir in the tomato paste and allow to slightly cook, about 1 to 2 minutes. Pour in vegetable broth while stirring and bring soup to a boil. Reduce heat and add the can of diced tomatoes and the potatoes. Simmer until potatoes are cooked through, about 20 minutes.
Meanwhile, preheat the peanut oil in an electric fryer or large pan fitted with deep-fry thermometer to 325 degrees F. In a large casserole dish, mix together flour, and season with salt and pepper. Dredge chops in flour mixture, patting excess off. Place chops, 2 at time, in fryer and cook until cooked through, about 6 minutes. Cool slightly and remove to a paper towel lined sheet tray.
Add the cooked chops to soup and simmer another 30 minutes, or until the meat falls off the bone. During the last 10 minutes of cooking, add the corn, peas, and green beans. Ladle soup into individual bowls, top with a pork chop and sprinkle with parsley.