Preheat oven to 400 degrees.
Cut the parchment paper into heart-shaped pieces, using the full width of the paper.
Rub one side of each heart with some oil. Place a snapper fillet on one half of each heart. Sprinkle with salt and pepper and drizzle each with half a tablespoon of wine. Combine tomatoes, garlic, thyme, capers and olives. Spread each fillet with one fourth of this mixture; top with a basil sprig. Fold other half of the paper heart over the fillet and tightly seal the package by making a series of tight overlapping folds along the open edge.
Place on baking sheet and bake until hearts puff up, about 20 minutes. Transfer hearts to individual plates and let each person open his or her package