In a large saucepan, combine half of the fish stock with the oil, vegetable broth and wine. Peel the potatoes and cut into 1 cm (approximately 1/2-inch) thick slices. Peel and squeeze the garlic through a garlic press into the saucepan. Bring to a boil and simmer about 20 minutes. Cut the red mullet into bite-size pieces, squeeze 1/2 of the lemon and drizzle the juice over the fish.
Rinse the mussels, remove the beards and cook the mussels in the saucepan until they open, discard any unopened mussels. Remove the meat from half the mussels, set the rest aside. Add the red mullet and the potatoes, and simmer over low heat until the potatoes are knife-tender, about 15 minutes. Drain the beans and add to the saucepan. Squeeze the other half lemon. Slice the whole lemon. Rinse the parsley, shake dry and chop finely.
Season the stew with the remaining lemon juice, paprika, salt and pepper, spoon into bowls and serve with lemon slices and parsley.