Bring a large pot of water to a boil; season generously with salt. Add celery root and cook, partially covered, 10 minutes. Add potatoes and parsnips and continue to cook, partially covered until vegetables are fork-tender, 15–20 minutes more.
Drain vegetables and pass through a ricer into now empty pot (or mash with a potato masher until smooth). Cook mashed vegetables over medium heat, stirring constantly, to remove extra moisture, 1–2 minutes.
Reduce heat to low and stir in butter until melted. Add milk and continue to cook, stirring frequently until silky and thickened, about 5 minutes. Stir in ½ teaspoon salt and ¼ teaspoon pepper; taste and adjust seasoning as needed.
Transfer parsnip mash to a serving dish, sprinkle with herbs, flaky sea salt, and black pepper; drizzle with melted butter or olive oil (optional).