Although it takes about an hour to prepare and fry enough chips for eight, it's worth the effort: They are absolutely irresistible. If kept in resealable plastic bags, they'll stay crisp for up to five days.
Shave long, thin strips from parsnips with a peeler (preferably swivel), turning to use all sides. (Reserve ends and cores for another use, such as stock or soup.) Toss parsnip strips by hand to separate.
Heat 3 inches oil in a heavy medium-size pot until it registers 365 degrees on a deep-fry thermometer. Add parsnip strips to the oil in small handfuls (do not crowd), turning gently with a wire skimmer, until pale gold and crisp, about 45 seconds. Transfer chips to paper towels to drain, and sprinkle with salt. (Return oil to 365 degrees between batches.) Before serving, sprinkle with smoked paprika.