Preheat an outdoor grill or grill pan over medium-high heat.
Brush both sides of the zucchini rounds with olive oil and season lightly with salt and pepper.
Grill for **3 to 4 minutes per side**, or until tender with light grill marks. Transfer to a serving platter and let cool slightly.
Meanwhile, in a medium bowl, combine the tuna, avocado, cherry tomatoes, parsley, lemon juice, and olive oil. Gently toss until everything is evenly coated. Season with salt and black pepper to taste.
Spoon the tuna mixture onto each grilled zucchini round.
Serve immediately.