Heat the olive oil in a large pot over medium-high heat. Add the sausage and cook, stirring often, until browned, about 5 minutes. Transfer to a plate using a slotted spoon.
Add the potatoes, wax beans, green beans, corn, tomato paste, thyme, 3/4 teaspoon salt and a few grinds of pepper to the pot and stir to coat. Add 4 cups water and the chicken broth and bring to a boil. Reduce the heat to medium low and simmer until the potatoes are tender, about 15 minutes.
Add the sausage to the pot and return the soup to a simmer; season with salt. Top each serving with the tomato, parsley and a drizzle of olive oil.