Wash the potatoes and cook in boiling water for 20-30 minutes, then drain and chill. In the meantime peel and dice the onion. Cut ham into cubes.
Peel and slice the potatoes. Heat 1 tablespoon of lard in 2 pans. Add 1/4 of each potato slices and fry for about 7 minutes while turning. Season with salt and pepper. After about 4 minutes add 1/4 of each onion.
Whisk 3 eggs twice and season with salt and pepper. Pour the eggs over the potatoes. Shortly before the mixture sticks, carefully push them together with a spatula. Allow to set to the end and turn out of the pan onto a plate.
Keep finished omelettes warm. Prepare 2 more omelettes from the remaining ingredients as described. Wash, clean and slice the tomato. Cut the cucumbers lengthwise in half and cut the halves fan-like.
Spread diced ham on the omelettes. Garnish with tomato, parsley and cucumber.