Fill a large pot with cool water and add the parsnips, sweet potatoes, rutabaga and garlic. Bring to a boil, then reduce the heat and simmer until the vegetables are very tender, about 20 minutes. Drain and return the vegetables to pot along with 4 tablespoons of the butter, the heavy cream, and salt and pepper to taste. Mash with a potato masher or a fork until combined. Keep warm.
In a small saucepan over medium heat, combine the remaining 4 tablespoons butter and the thyme sprigs. Cook until the butter is toasted and brown specks start to appear, about 3 minutes.
To serve, spoon the mashed vegetables into a serving dish, top with thyme sprigs and drizzle the brown butter over the top.