Heat the oil in a large pot over medium-high heat. Add the onion to the pot and sauté for a couple of minutes, until softened a bit. Add the oregano and garlic; sauté 1 minute. Stir in the squash, zucchini and corn, and continue to sauté a few more minutes until veggies begin to get tender. Remove from heat.
Place 1 can of (drained) tomatoes and 1 can of broth in a blender and process until smooth. Pour into the pot with the vegetables and return to heat. Stir in the second can of (drained) tomatoes and 2 cans of broth. Bring the mixture to a boil; reduce heat and simmer for about 20 minutes.
Add the beans and Swiss chard to the pot and stir until the chard begins to wilt (about 5 minutes). Remove from heat and stir in salt and pepper.
Ladle the soup into bowls and sprinkle Parmesan on top of each individual serving.