Season the fish filets with salt, pepper and cumin. In a Dutch oven, heat the oil over medium-high heat. Add the fish and cook until golden on both sides and cooked through. Remove the filets from the Dutch oven and set aside.
In the same pot, add the chopped vegetables and cook until soft, about 5 minutes. Season the refrito with salt, pepper, cumin, aji powder and cayenne pepper. Stir to incorporate. Add four cups of water and bring the liquid to a boil. Upon boiling, remove the soup from the heat.
Allow the liquid to cool slightly. Puree the vegetables with the liquid using a blender. Strain back into the Dutch oven. Add 1 cup of water, followed by the diced potatoes, and cook over medium heat until the potatoes are tender. Taste and adjust the seasoning, if necessary.
While the soup simmers, pickle the onions. Toss the sliced onion with the lemon juice, salt and sugar. Stir and let rest until the onions are soft, about 15 minutes .
To serve, flake the fish and mix with the pickled onions. Ladle some of the potatoes and soup into a bowl. Top with the fish/pickled onion mixture and garnish with chopped parsley.