Heat peanut oil in a 5-quart Dutch oven over high heat to between 370 and 375ºF.
Meanwhile, scrub the parsnips to remove any excess dirt. Lay the parsnip flat on a cutting board, and using a vegetable peeler, peel off 4 to 5 flat, wide noodle-shaped strips. Rotate the parsnip 90 degrees and repeat. Continue to rotate until you can no longer peel strips. Repeat with the remaining parsnips.
Gently add a small handful of parsnips to the oil, stirring gently, until lightly browned and crisp, 1 to 1 1/2 minutes. Remove the crisps from the oil using a spider or slotted spoon; hold over the pot and allow to drain for 30 seconds. Transfer to a wire rack set in a half-sheet pan and sprinkle with salt and pepper, if desired. After the first batch, the oil temperature may drop. Adjust the heat in order to maintain a minimum temperature of 325 to 350ºF. Repeat until all parsnips are cooked. Serve warm or at room temperature. Once completely cooled, store in an airtight container for up to 3 days.