Make this salad just before serving to avoid extracting excess moisture from the celeriac and producing a watery dressing. Celeriac is also sold as celery root or celery knob. Use a mandoline to produce paper-thin slices.
In a small bowl, combine mayonnaise, cornichons, capers, and mustard. Season with salt and pepper. Blend until well combined. Fold in tarragon and parsley; set aside.
Slice celeriac, using a Japanese mandoline fitted with the julienne blade, into 1/8-inch-thick strips into a large bowl. Add mayonnaise mixture, and toss to combine. Season to taste with salt and pepper.