In a medium bowl, combine the drained tuna, diced cucumber, diced avocado, red onion, and celery.
In a small bowl, mix together the mayonnaise, lemon juice, salt, and pepper until well combined.
Pour the mayonnaise mixture over the tuna and vegetable mixture. Gently fold everything together until well coated.
Taste and adjust seasoning if necessary. If desired, chill the salad in the refrigerator for 15-30 minutes before serving to enhance the flavors.
Serve the salad on a bed of greens or in a sandwich, and garnish with fresh parsley.