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.
Combine the vinegar, lemon juice, honey, salt, and pepper in a blender. With the machine running, gradually add the oil and blend until emulsified.