Slice pork loin into 1/4-inch slices and lightly pound out until thin. Season pork slices with salt and pepper on both sides, and then dredge slices in flour, shaking off the excess.
Heat 4 tablespoons butter and the olive oil in large sauté pan over medium-high heat until hot. Sauté the pork slices for 3 minutes and flip, cooking the other side for 3 minutes, making sure that each side is nicely browned.
Remove pork to a plate and deglaze the pan with lemon juice, chicken stock, and capers, using a wooden spoon to scrape the bottom of the pan. Cook until the liquid reduces by about 1/3.
Season with salt and pepper, and stir in the remaining 2 tablespoons of butter to slightly thicken. Place pork on platter or plates and garnish with the lemon-caper sauce and parsley to serve.