Heat peanut oil in a 5-quart Dutch oven over high heat to 370 to 375 degrees F.
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, approximately 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 cooling rack set over 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 350 degrees F. 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.