Place the cod in enough cold water to cover and let sit, changing the water periodically, for 24 to 36 hours.
Drain the cod and discard the soaking water. Cut the cod into 2 or 3 pieces and place in a saucepan with the potato. Add enough water to cover and bring to a boil. Simmer, covered, until the potato and cod are both tender, about 20 minutes. Remove the potato and salt cod and reserve the cooking liquid separately. Transfer the cod to a food processor and process, adding 3 to 4 tablespoons of the cooking liquid, as needed, to form a smooth paste. Set aside.
Preheat the oven to 350 degrees F.
Pass the potato through a ricer or strainer into a mixing bowl while it is still hot. Heat 6 tablespoons of the olive oil gently in a small saucepan and, when warmed, gradually stir the oil into the potato. Add 3/4 teaspoon of the garlic and the reserved cod mixture and stir to combine. Season, to taste, with salt and freshly ground white pepper. Transfer the spread to a small bowl and sprinkle with the parsley. Set aside while you prepare the toasts.
Combine the remaining 1/2 cup of olive oil and remaining 3/4 teaspoon of garlic in a small cup or ramekin and, using a pastry brush, lightly coat the bread slices with some of the oil mixture on 1 side. Season lightly with salt and white pepper. Transfer to a baking sheet and bake until lightly golden and crisp, 10 to 12 minutes. Serve the toasts with the warm cod spread, allowing guests to assemble their toasts so that they are eaten immediately, with about 1 tablespoon of the cod spread spooned onto each toast. (If toasts are assembled with spread ahead of time the toasts will become soggy.)
Note: This is best served warm or at room temperature.