Using sharp knife, cut flesh off skin, then discard skin. Chop salmon flesh into 1/4- to 1/3-inch pieces and mix with chopped bread, mayonnaise, onion, parsley, salt, and lemon juice in medium bowl. Scoop a generous 1/4-cup portion salmon mixture from bowl and use hands to form into a patty measuring roughly 2 1/2-inches in diameter and 3/4-inch thick; place on parchment-lined baking sheet and repeat with remaining salmon mixture until you have 8 patties. Place patties in freezer until surface moisture has evaporated, about 15 minutes.
Meanwhile, spread flour in pie plate or shallow baking dish. Beat eggs with 1 1/2 teaspoons vegetable oil and 1 1/2 teaspoons water in second pie plate or shallow baking dish, and spread bread crumbs in a third. Dip chilled salmon patties in flour to cover; shake off excess. Transfer to beaten egg and, using slotted spatula, turn to coat; let excess drip off. Transfer to bread crumbs; shake pan to coat patties completely. Return now-breaded patties to baking sheet. They can be frozen at this point, then bagged together to cook from frozen when you want them. Double cooking time if frozen.
Heat remaining 1/2 cup vegetable oil in large, heavy-bottomed skillet over medium-high heat until shimmering but not smoking, about 3 minutes; add salmon patties and cook until medium golden brown, about 2 minutes. Flip cakes over and continue cooking until medium golden brown on second side, about 2 minutes longer. Transfer cakes to plate lined with paper towels to absorb excess oil on surface, if desired, about 30 seconds, and then serve immediately.