Preheat oven to 450°. Line 2 rimmed baking pans with parchment paper. On mandolin with largest julienne blade, cut parsnips lengthwise.
In large bowl, toss oil, salt and parsnips until coated. Transfer parsnips to prepared baking pans. Bake fries 25 minutes or until golden brown and crisp, turning once halfway through cooking. Sprinkle with parsley; serve with aioli for dipping.