Rinse bacalao to wash off as much salt as possible. Cut tayota into quarters. Boil bacalao and tayota in a gallon [4 l] of water until the tayota is tender (20 - 25 mins), adding water as it becomes necessary to keep the same water level. Remove from the heat and discard water. Cool to room temperature. Taste the bacalao, if it still is too salty, submerge in clean water and let it rest for half an hour.
Heat the oil over medium-low heat in a large pan. Add onion and cook and stir until it starts to become translucent. Add garlic cook and stir for 30 seconds. Add bell pepper and tomato. Cook and stir for 2 minutes. Add tayota and bacalao, cook and stir until everything is well mixed. Pour in eggs, cook stirring until the eggs set. Season with salt and pepper to taste. Sprinkle with parsley.