Using the mandolin, shave the apples as thinly as possible into your bowl. Next, the fennel: Holding it by the root end, shave the fennel crosswise into the mixing bowl. Toss in the parsley leaves and walnuts. Slice the lemons in half and squeeze them over the salad. Add a splash each of olive oil and walnut oil. Toss well and season with salt and pepper to taste.
*A note on walnuts and walnut oil: Do your best to find a walnut purveyor at your local farmers’ market, and make sure the oil and nuts are fresh. It really makes all the difference.