Put the sweet potato, baking potato and parsnip wedges in a bowl and drizzle with the olive oil, tossing well to coat lightly. Sprinkle with the Cajun seasoning and toss again to coat. Transfer to a large baking sheet and roast in a preheated oven, 200°C (400°F), Gas Mark 6, for 25–30 minutes, until the vegetables are crisp and golden.
Meanwhile, make the mayonnaise. Place all the ingredients except the parsley in a small measuring jug, and using an electric blender whiz until a thick mayonnaise is formed. Stir in the parsley.
Serve the chips tossed with the parsley, with a tub of the mayonnaise, to dip.