Warm the oil in a 5-quart heavy-duty pot over medium-low heat. When it’s warm—not hot—add the celery, carrots, garlic, onion, and parsley. Cook, stirring occasionally, until the onion is translucent and the carrots have begun to soften, about 10 minutes.
Stir in the eggplant, potato, yellow squash, zucchini, 1/2 tsp. salt, and a few grinds of pepper. Cook, stirring often (the potato tends to stick to the bottom of the pot), until the vegetables are tender but still hold their shape, 15 to 20 minutes.
Add the corn and 6 cups of the broth; bring to a boil. Turn the heat down to low, partially cover, and simmer gently for 10 minutes. Return to a boil and stir in the pasta. Simmer, stirring once or twice, until the pasta is al dente or even a little bit more tender; cooking time will depend on the shape and brand of pasta you use. Add more broth to thin the soup, if you like. Season to taste with salt and pepper.
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