Heat 1 tablespoon oil in a medium saucepan over medium heat. Add parsnips with ½ teaspoon salt, cook for for 5 minutes, stirring frequently.
Add potatoes and 1 teaspoon salt to the pot, cover with water. Bring to a boil., then reduce heat, and simmer until parsnips and potatoes are tender, 12 to 15 minutes.
Drain parsnips and potatoes. Return empty pot to medium-high heat, make sure any remaining water evaporates before adding oil. When pot is dry, add remaining 4 tablespoons oil and the thyme leaves. Be very careful adding thyme as it will sometimes pop when it hits the hot oil.
Cook until thyme starts to sizzle, about 1 minute. Add parsnips and potatoes, and mash with a potato masher or spoon until combined, but still chunky.
Season to taste with salt and pepper. Drizzle with more oil and garnish with more thyme leaves when serving.