In a small bowl, stir together the garlic, parsley, salt and pepper. Set aside. Peel and cube the potatoes into 1-inch pieces.
Remove the outer leaves and stem of the artichokes. Gently ease the center of the artichoke open. Spoon half the garlic mix into the opened center of one artichoke. Repeat with the second artichoke. Stand the artichokes up in a pot, a size that will hold both tightly. Typically I use a 7-inch wide pot, but this will depend on the size of your artichokes. Add the cubed potatoes. Drizzle the olive oil all over. Add water until it reaches halfway up the artichokes (about 2-3 cups).
Cover the pot and bring to a boil over medium-high heat. Once the water comes to a boil, reduce the heat to low, place the lid ajar and leave to simmer for 30-35 minutes. The artichokes are cooked when you can easily remove an inner leaf and the base of the leaf is soft enough to eat. Transfer the artichokes and potatoes to your serving dish and spoon over some of the cooking liquid. Drizzle with more olive oil and a sprinkle of salt, if desired. Enjoy while warm.
Once you have peeled off the leaves and eaten the meat in them, don’t forget about the base of the artichoke. Remove the hairy choke and at the bottom of it all you should find beautifully cooked base that can be eaten with a knife and fork.