Preheat oven to 350°. Lay one sheet filo dough on a work surface and brush generously with melted butter. Lay another sheet on top, brush with butter, and repeat with remaining sheets.
Using a pizza cutter or very sharp knife, cut dough into 12 equal rectangles. Place each in the well of a mini muffin pan, gently pressing into the bottom and sides.
Bake filo cups until golden brown, 8 to 9 minutes. Let cool on a cooling rack, then remove each filo cup from pan.
Fill filo cups evenly with caponata. Sprinkle with feta cheese, pine nuts, and parsley and serve immediately.