Combine potatoes and water in a pot and bring to a boil, heavily season with salt, cook until very tender, strain and place back into the pot. Add yogurt, butter, salt, and pepper, and mash with a potato masher until smooth.
Preheat the oven to 400°F.
In an oven-safe skillet, heat the olive oil on medium heat. Add the onions and cook until golden, about 5-7 minutes. Reduce heat to low, add mushrooms, carrots, garlic, and thyme, and cook until the vegetables soften, about 10 minutes.
Add quinoa, tomato paste, and salt, cook until the tomato paste begins to stick to the vegetables, 3 more minutes. Then add the vegetable broth and peas and stir to combine. Carefully place mashed potatoes on top of the filling and spread into an even layer.
Place the pie in the oven and bake until the potatoes begin to brown, about 25-30 minutes, then broil for 2-3 minutes until the potatoes become golden. Remove from the oven and let stand for 10 minutes. Sprinkle with chopped parsley and serve.