Put the potatoes in a large saucepan with enough water to cover by at least 1 inch. Add 1 Tbs. salt. Bring to a boil, reduce to a rapid simmer, and cook until the potatoes are tender all the way through, 5 to 8 minutes. Drain well in a colander and let sit.
Meanwhile, in a large nonstick skillet over medium-high heat, heat 1 Tbs. of the butter and 1 Tbs. of the olive oil. When the butter is melted and the pan is hot, add the onion and green pepper and sauté, stirring frequently, until the vegetables are soft and the onions are well browned, about 8 minutes. Push the vegetables to one side and add the ham. Leave the ham alone in the pan for a few minutes. When it begins to turn crisp, add the garlic and stir all the ingredients together. Continue cooking until the garlic is fragrant and begins to soften, 1 to 2 minutes. Transfer the mixture to a bowl.
Heat the remaining 1 Tbs. butter and 1 Tbs. olive oil in the pan over medium-high heat. Add the drained potatoes, distributing them in one layer if possible. Let the potatoes sit undisturbed to brown, about 5 minutes. Meanwhile, sprinkle 1/2 tsp. salt over the potatoes. Return the onion and pepper mixture to the pan and mix it into the potatoes. Cook the hash for another 3 to 5 minutes. As it cooks, alternately fold the mixture over onto itself and mash it together with a metal spatula, letting it sit for periods of about 30 seconds in the pan to brown.
Mash the hash just enough so that it holds together but the potatoes are still in discernible pieces. Grind a generous amount of fresh black pepper (8 to 10 grinds) over the hash and fold it in. Remove the pan from the heat, add the parsley, and serve.