Melt the butter in a frypan over medium heat. Stir in the onion and cook until the onion has softened and turned translucent, about 5 minutes.
Transfer the onion to a mixing bowl and stir in the tuna, parsley, capers and Parmesan cheese; season to taste with salt and pepper.
Cut the filo sheets in half lengthways, ending up with long, narrow sheets. Lay out one strip of filo at a time on your work surface with one of the narrow ends close to you. Place a heaping spoon of filling 3cm from the end closest to you. Fold the bottom right corner over the filling to the left edge to form a triangle. Fold the triangle up, bringing the point at the bottom left up to rest along the left edge. Turn the lower left corner over to touch the right edge. Continue turning the triangle over in this manner until you reach the end of the filo. Lightly brush the last 3cm with water to seal. Repeat with the remaining filling and dough.
Heat oil in a deep-fryer or large saucepan to hot (180 degrees C).
Fry the triangles in the preheated oil in batches until golden brown and crispy, about 3 minutes. Drain on a paper towel-lined plate. Serve with lemon wedges.