Cut the parsnips. Peel the parsnips and cut them into 2-inch (5 cm) pieces. Quarter any thick pieces from the top of the root and cut out the woody core, which stays stringy after cooking.
Steam the parsnips. Pour about 2 inches (5 cm) of water into a large pot, set a steamer basket or rack inside, and bring the water to a boil. Add the parsnips, cover, and steam for about 10 minutes, until a thin knife slides into the thickest piece with no resistance.
Prep the squid while the parsnips steam. Pat the squid dry. Slit the tubes open and lay them flat with the inside facing up. Score a shallow crosshatch about 1/2 inch (1 cm) apart, cutting only partway through, then cut the tubes into 1 1/2-inch (4 cm) pieces. Leave the tentacles whole, or halve any large clusters.
Season the squid. Toss the squid with the salt, pepper and sesame oil and let it stand for 5 minutes.
Make the purée. Transfer the parsnips to a food processor, leaving the water in the pot on low heat. Scoop out 1/4 cup (60 ml) of the hot water, add it a tablespoon at a time along with the butter and salt, and blend for about 1 minute, until smooth and thick enough to hold a spoon upright. Mash with a potato masher for a rustic texture. Cover to keep warm.
Plate the squid. Spread half of the scallion and ginger on a shallow heatproof plate that fits inside the pot. Pile the squid on top in no more than two layers, then scatter the rest of the scallion and ginger over it.
Steam the squid. Bring the water back to a rolling boil. Lower in the plate, cover, and steam for 4 to 5 minutes. Check at 4 minutes: the squid should look opaque white with curled edges and feel springy, not stiff. Lift the plate out with tongs or oven mitts as soon as it is ready.
Add the hot oil. Heat the neutral oil in a small pan for 1 to 2 minutes, until it shimmers and a scallion strip sizzles when dropped in. Pour it over the squid and scallions, which should hiss and release a strong ginger aroma. Add the soy sauce, rice vinegar and chili, if using.
Serve. Spoon the purée onto warm plates, top with the squid, and drizzle the juices from the steaming plate over both. Serve right away.