Drizzle the corn with 2 tablespoons olive oil and sprinkle it with salt and pepper. Add the corn to the grill and cook, turning occasionally, until the corn starts to char, 6 to 8 minutes. Set the corn aside to cool slightly.
In a large bowl, whisk the remaining 3 tablespoons olive oil with the capers, chopped parsley, vinegar and mustard. Sprinkle with salt and pepper. Add the farro, mozzarella, tomatoes, olives and cucumber and toss to coat with the dressing. Cut the corn kernels from the cob. Fold the corn and avocado into the farro mixture. Sprinkle with the parsley leaves. Serve immediately, or chill until ready to serve.