Preheat oven to 450 degrees F. Cut 4 large sheets of parchment paper, each about 16 by 12 inches (or use pre-cut parchment sheets).
Combine mayonnaise, parsley, mustard, honey and turmeric in a small bowl.
To make packets, lay the parchment sheets on a work surface with the long sides closest to you. Fold each in half (bringing the short sides together), then open. Place 1/2 cup potatoes on one side of each piece of parchment and sprinkle with 1/4 teaspoon salt and 1/8 teaspoon pepper. Top each with 1 cup of kale and a piece of tuna. Sprinkle the tuna with the remaining 1/4 teaspoon each salt and pepper. Brush with the mayonnaise mixture. Close the packets and seal the edges with small, tight folds. Place the packets on a large baking sheet.
Bake until the fish is just cooked through, 10 to 15 minutes. (Carefully open one packet to check for doneness–be cautious of the steam.) Set each packet on its own plate. Let stand for 3 minutes. Cut an X in the top of each packet with scissors and carefully fold open to serve.