Add the olive oil to a large skillet over medium-low heat, followed by the onions, peppers and garlic. Cook, stirring gently from time to time, until soft, about 20 minutes. Stir in the cumin, smoked paprika, piment d'Espelette, salt and pepper. Stir in the tomatoes and chopped parsley, reserving a handful for garnish, and let the mixture simmer about 5 minutes.
Using the back of a spoon, create a crater in the tomato mixture and gently crack an egg into the crater; repeat with the remaining 5 to 7 eggs. Continue cooking on the stovetop until the whites are set but the yolks are still runny, 4 to 6 minutes. Alternately, transfer the skillet to the oven for 5 minutes until the whites are set and the yolks are still runny. Garnish with reserved parsley.