Preheat the oven to 425°F (220°C). Toss the rutabaga with 3 tablespoons olive oil, salt, and pepper, and spread on a sheet pan.
Roast the rutabaga alone for 25 minutes, stirring once halfway through.
While roasting, simmer the apple cider with thyme sprigs until reduced by half and thickened, 12 to 15 minutes. Remove the thyme, stir in the vinegar, and set aside.
Pat the anchovies dry and toss with the remaining olive oil, salt, and pepper.
Push the rutabaga to one side of the pan and arrange the anchovies on the other, skin-side up. Roast 10 to 12 minutes until the anchovies are cooked through and the rutabaga is fork-tender.
Remove from the oven and brush the reduced apple cider glaze over both.
Scatter fresh parsley over the top and serve directly from the pan.