Heat the olive oil in a large Dutch oven over medium heat. Add the diced onion and cook for 5 minutes, stirring occasionally, until soft and translucent.
Add the minced garlic and cook for 1 minute, stirring constantly, until fragrant but not browned.
Add the carrots and celery. Cook for 3 minutes, stirring once or twice, until edges begin to soften. Season with ½ tsp salt and the black pepper.
Add the cubed potatoes, canned diced tomatoes with their liquid, and vegetable broth. Add thyme, oregano, and bay leaves. Stir to combine.
Raise heat to medium-high and bring to a full boil. Reduce to medium-low and simmer uncovered for 15 minutes, until potatoes are nearly tender when pierced with a knife.
Add zucchini and green beans. Simmer for 10 minutes, until green beans are tender with a slight bite and zucchini is cooked through but firm.
Remove bay leaves. Taste and adjust salt as needed. Ladle into bowls, garnish with fresh parsley, and serve immediately.