Make the Lyonnaise Garnish: Heat the olive oil in a large nonstick skillet over medium-high heat. Add the potatoes and cook, tossing, for 10 minutes. Add the onions, season with salt and pepper, and cook until the onions are translucent and the potatoes are tender, 5 to 10 minutes. Pour in the vinegar and cook until it has reduced to a glaze.
Make the Seared Cod: Heat the olive oil in a nonstick skillet over medium-high heat. Season the fillets with salt and pepper and slip them into the pan, skin side down, along with the garlic and thyme. Sear the fillets for 3 minutes, turn them over and cook for 3 minutes more. Reduce the heat to medium, add the butter and cook another 3 minutes, or until the fish is opaque, moist and lightly firm when pressed.
Spoon the Lyonnaise garnish onto a serving platter, top with the seared cod, and sprinkle with parsley.