Add the oil to a large cast-iron skillet over medium heat, then line the bottom with the potato rounds. Season with salt and pepper, add enough water to cover and bring to a boil. Cook until softened, about 5 minutes. Once the potatoes are softened, drain off the excess water so there is only a small amount left on the bottom. Scatter the lemons, shallots, tomatoes, wine, olives, capers and thyme over the potatoes. Dot with the butter. Season the fish with salt and pepper on both sides and put on top of the potatoes.
Make a parchment lid for the skillet by cutting a piece of parchment paper that's larger than your skillet. Fold it in half, then fold it in half again in the other direction so you have creases on 2 sides of the paper. Starting with the folded tip, fold on the diagonal 3 times, like a fan, to form narrow triangles. Measure how large your parchment lid should be by placing the tip of the triangle above the center of the skillet and cutting it where it hits the edge of the skillet. Cut off the folded tip to make a small hole in the center. Open up the parchment and set aside.
Open the parchment lid and place it over the skillet. Transfer the skillet to the oven and bake until the potatoes are soft and the halibut is fully cooked and no longer translucent, 15 to 20 minutes.
Remove the parchment lid. Serve, drizzled with some of the sauce from the skillet and some oil, then sprinkle with the parsley.