Place parsnips and carrots in a large pot; add enough salted water to cover. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until vegetables are tender, 15 to 20 minutes. Drain; return vegetables to the pot. Reduce heat to low; stir in 3 tablespoons butter and chives.

Purée vegetables using an immersion blender until nearly smooth; add remaining 3 tablespoons butter and continue to purée until completely smooth. Season with salt and black pepper.