In a large pot over medium-high heat, add olive oil. When oil shimmers and ripples, add onions, celery, carrots, squash, potatoes, zucchini, wax beans, salt, pepper, and thyme. Cook until veggies become slightly tender, about 5 minutes.
To a medium-sized bowl, add tomato paste, and about 1 cup of the water. Whisk until completely combined and not lumpy, and then add it to the vegetable mixture.
Add remaining water and canned beans with liquid. Turn heat up to high. Bring mixture to a boil, and then turn heat down to low and simmer for 20 minutes, or until vegetables are tender and potatoes can be easily pierced with a fork.
Turn heat off and mix in parsley. Divide among bowls and serve! Garnish with more parsley leaves.