Preheat an oven to 425 degrees F. Toss the onions, carrots, celery, turnip, bell pepper, sweet potato, mushrooms, and garlic with olive oil on a large rimmed sheet pan. Spread out into an even layer and roast for 30 minutes, or until the vegetables are tender and lightly browned at the edges.
Scrape the vegetables into a 5- to 6-quart slow cooker. Pour 1 cup of the vegetable broth onto the sheet pan and scrape up any browned bits clinging to it; scrape into the cooker. Add the remaining 5 cups broth, the diced tomatoes and their liquid, salt, pepper, rosemary, and thyme, cover the cooker and cook for 2 to 3 hours on high, or 4 to 6 hours on low, until the flavors are blended.
Stir in the couscous and parsley and cook for 5 minutes more.