For the potatoes: Preheat a grill pan over medium-high heat. Place the potato slices in a large bowl and toss with the olive oil, salt and pepper. In batches, grill the potatoes until cooked through and golden, about 4 minutes per side. Transfer to a large serving bowl and set aside to cool while you make the dressing.
For the dressing: In a small bowl, whisk together the lemon juice, lemon zest, olive oil and Parmesan. Pour the dressing over the potatoes, then add the basil and parsley, and toss gently to coat. Garnish with shaved Parmesan, and serve.