Cut vegetables separately into 2-in. chunks. Put each in a separate saucepan, add enough cold water to barely cover, and add 1 1/2 tsp. salt to potatoes and 1 tsp. to celery root. Bring to a boil over high heat, then lower heat and simmer until tender, about 15 minutes. Transfer vegetables to a large bowl and reserve their cooking liquids.
While vegetables cook, simmer cream and garlic in a small pot, covered, until garlic is tender, about 10 minutes.
Using a potato masher, mash vegetables until smooth, drizzling in garlic cream and enough celery root cooking liquid to make it as loose as you like (if you need more, use potato cooking liquid). Mix in chopped parsley and 2 1/2 tbsp. hazelnuts. Season to taste with salt and pepper.
Swirl in hazelnut oil and sprinkle with remaining 1/2 tbsp. hazelnuts and the parsley leaves. Serve hot.