Preheat oven to 375. Line a cookie sheet with foil. Brush cut side of squash and garlic cloves with oil. season with salt and pepper. Place the squash cut sides down on the sheet, tucking the garlic cloves and thyme into the squash cavities. Roast about 45 minutes.
Meanwhile, place the potato in a medium saucepan and fill with enough water to cover by 1 inch. Bring to a low boil and cook until tender, about 15 minutes Drain and set aside.
When the squash is cool enough to handle, remove and discard the peel and the thyme. Cut the squash into large pieces. In a blender, working in batches, puree the squash, garlic, potato and chicken broth. Season with salt and pepper.
In a medium saucepan, heat the soup over medium-low heat until warmed through. Divide among 4 bowls and top with yogurt, onions and parsley.