Toss the chicken, bell peppers, onion, parsley, almonds, and capers in a large bowl with enough vinaigrette to moisten. Season the chicken salad with salt and pepper, to taste.
Arrange 1 large lettuce cup and 1 small lettuce cup on each plate, overlapping slightly. Spoon the chicken salad into the lettuce cups. Drizzle more vinaigrette over the salads. Sprinkle with the Parmesan and serve.
Do-Ahead Tip: The chicken salad can be prepared 4 hours ahead and refrigerated. Just before serving, spoon the salad into the lettuce cups.
Red Wine Vinaigrette:
1/2 cup red wine vinegar
1/4 cup lemon juice
2 teaspoons honey
2 teaspoons salt
Freshly ground black pepper
1 cup olive oil
Combine the vinegar, lemon juice, honey, salt, and pepper in a blender. With the machine running, gradually add the oil and blend until emulsified.