In a bowl, whisk the yogurt with the 1/4 cup of parsley and season with salt and pepper. Cover and refrigerate until chilled.
In a large enameled cast-iron casserole, heat the olive oil. Add the onion, garlic and a generous pinch each of salt and pepper and cook over moderate heat, stirring occasionally, until softened and just starting to brown, about 8 minutes. Stir in the tomato paste, cumin, coriander, crushed red pepper, caraway and cinnamon and cook, stirring, until fragrant and glossy, about 3 minutes.
Add the turnips, parsnips, tomatoes and their liquid, broth, saffron and 1 cup of water to the pot and bring to a boil. Cover and simmer over moderately low heat, stirring occasionally, until the turnips and parsnips are beginning to soften, about 20 minutes.
Add the sweet potatoes, cauliflower, chickpeas and currants to the pot. Cover partially and simmer over moderately low heat, stirring occasionally, until the vegetables are tender, about 20 minutes. Stir in the olives and season with salt and pepper. Transfer the tagine to a serving bowl, garnish with parsley and serve with the yogurt.