Stir bread crumbs and vinegar together in a small bowl and let sit 5 minutes.
Pound anchovies, garlic, and 4 tbsp. capers into a paste in a mortar. Add egg yolks and tomato paste, then parsley, and pound to mix. Pound in bread crumbs. Stir in enough oil to make a thick sauce. If using a food processor, whirl soaked bread crumbs with anchovies, garlic, 4 tbsp. capers, the yolks, tomato paste, and parsley; then drizzle in the oil.
Turn into a bowl and top with remaining 2 tbsp. capers.
Make ahead: Up to 1 week, chilled, with plastic wrap pressed against surface.