Heat 1 tablespoon oil in a medium saucepan over medium heat. Cook parsnips with 1/4 teaspoon salt for 5 minutes. Add potatoes and 1 1/2 teaspoons salt; cover with water. Bring to a boil. Reduce heat, and simmer until parsnips and potatoes are very tender but still hold their shape, 12 to 14 minutes.
Drain parsnips and potatoes. Return empty saucepan to high heat, allowing any remaining water to evaporate. Add remaining 3 tablespoons oil and the thyme. Cook until thyme starts to sizzle, about 1 minute. Add parsnips and potatoes, and smash with the back of a wooden spoon until combined but still chunky. Season with salt and pepper. Drizzle with oil.
Smashed potatoes and parsnips can be refrigerated for up to 2 days. Warm over low heat, stirring occasionally, before serving.