Cook the potatoes in a large saucepan of salted boiling water for 10 minutes. Add the carrots and cook for a further 3 minutes, then add the peas and continue to cook for 2 minutes until all the vegetables are just cooked through. Drain, then cool under cold running water and drain again.
Mix together the mayonnaise and mustard in a large bowl, then add most of the spring onion, cornichons, capers and parsley, reserving some for garnish. Stir in the cooled vegetables, then season with salt and pepper to taste.
Transfer the salad to a serving bowl and serve scattered with the reserved ingredients.