Drain both cans of tuna well in a colander and transfer to a mixing bowl, flaking with a fork.
Add the grated garlic, finely chopped onion, salt, black pepper, lemon juice, lemon zest, egg, 1/4 cup breadcrumbs, 2 tablespoons mayonnaise, 1/4 cup grated Parmesan, and dried parsley to the bowl; mix until combined.
In a shallow dish, combine the remaining 4 tablespoons breadcrumbs and 3 tablespoons grated Parmesan for the coating.
Scoop about 1/4 cup of the tuna mixture and shape into a patty; repeat to make 8 cakes.
Lightly coat each patty in the breadcrumb-Parmesan mixture, pressing gently so it adheres.
Heat 1 to 2 tablespoons of olive oil in a frying pan over medium heat; add cakes (work in batches if needed) and cook until golden brown, about 3–4 minutes per side.
Transfer cooked tuna cakes to a plate and let rest 5 minutes before serving.