Peel, wash, and cut potatoes in medium-sized cubes. Place them in a bowl of water. Peel, wash, and cut carrots in half-moons. Wash cauliflower and separate it into florets.
Fill a pot with water, add salt and the carrots, and bring it to a boil over medium-high heat. Once boiling, cook the carrots for 10 minutes.
After 10 minutes, drain the potatoes and add them to the pot with the carrots. Cook for 15 minutes, then add the cauliflower florets and cook for another 7 minutes. Drain the vegetables and transfer them to a large mixing bowl.
Preheat the oven to 356°F (180°C).
In a bowl whisk together yogurt, sour cream, parsley flakes, and salt. Taste and add more salt if necessary. Then whisk in the flour.
Pour about ¾ of the yogurt sauce over the vegetables and gently toss to coat. Transfer it to the baking dish, and pour the remaining yogurt sauce on top.
Cover the baking dish with aluminum foil and bake for 25 minutes. While the casserole is baking, grate the cheese.
After 25 minutes, remove the foil, and top the casserole with the grated cheese. Set your oven to broiler settings, and broil the casserole for a few more minutes, until golden brown.