Bring broth and wild rice to a boil in a medium saucepan. Reduce the heat to an active simmer, cover, and cook for 45 minutes. Drain and set aside.
In the meantime, in a large skillet, heat the olive oil and butter over medium-high heat. Add the mushrooms, garlic and thyme, and sauté until soft and golden, about 6-7 minutes. Season to taste with salt and pepper.
Deglaze the skillet with the white wine and let evaporate 2 minutes. Remove thyme sprig and discard. Add the shallots and wild rice, and mix well until heated through, adding broth by the tablespoon, if needed. Remove to a serving platter and sprinkle with the fresh parsley.