Toss the potatoes, onion and garlic with 1 tablespoon olive oil, the red pepper flakes and 1/4 teaspoon salt in a bowl. Put the cod on a plate; drizzle with the lemon juice and season with salt, pepper and the thyme.
Preheat the oven to 425 degrees F. Tear off four 16-inch sheets of parchment paper. Make the packets (see photo gallery): Arrange about one-quarter of the potato mixture on one sheet of parchment; top with a cod fillet and about one-quarter each of the olives and tomatoes (along with some liquid from the can). Drizzle with 1/2 tablespoon olive oil. Seal as directed. Repeat to make 3 more packets; divide between 2 baking sheets.
Transfer to the oven and bake until the parchment puffs and the liquid is bubbling, about 25 minutes. Let rest 5 minutes before opening.
Meanwhile, mix the butter, chives, lemon zest and a pinch of salt in a small bowl. Carefully open the packets; top with the chive butter.