Bring a large pot of salted water to a boil. Add rice, cover, reduce heat and cook until tender, about 30 minutes. Drain and rinse under cold running water. Set aside.
Warm oil in a large pot over medium-high heat. Add carrots, onion and celery and cook, stirring, until slightly softened, about 3 minutes. Add garlic; cook and stir 1 minute. Add chicken, then broth. Bring to a boil over high heat, then reduce to low. Simmer for about 10 minutes.
Add cauliflower, broccoli and peas to soup. Stir in rice. Cook until vegetables are soft, about 10 minutes. Sprinkle with parsley; season with salt and pepper. Serve hot.