Place the drained chickpeas in a large bowl. Using a fork or potato masher, gently mash about half of the chickpeas so you have a mix of mashed and whole beans for a chunky texture.
Add the flaked tuna, diced red pepper, chopped red onion, and parsley to the bowl with the chickpeas and fold together until evenly combined.
In a separate small bowl, whisk the mayonnaise, Greek yogurt, olive oil, lemon juice, lemon zest, and Dijon mustard until smooth. Stir in the chopped capers, salt, pepper, and red pepper flakes if using.
Pour the dressing over the chickpea-tuna mixture. Use a spatula to gently fold everything together, taking care not to overwork the tuna so it remains flaky. Taste and adjust seasoning with additional salt, pepper, or lemon juice as needed.
Cover and chill the salad for at least 10 minutes to let the flavors marry (this step is optional but recommended). Stir once more before serving.
Serve the salad over mixed greens, scooped into pita or on toasted bread for sandwiches, or enjoy by itself with a sprinkle of extra parsley and a drizzle of olive oil.