Heat the oil in a large skillet over medium-high heat. Add the parsnips and sprinkle with salt and pepper. Cook, turning occasionally, 6 to 8 minutes. Then add the carrots and continue cooking until the vegetables begin to brown and soften slightly, another 10 minutes.
Add the butter and thyme to the vegetables and continue cooking until the vegetables are glazed, about 3 minutes. Season with salt and pepper, sprinkle with the parsley and serve warm.