Heat the olive oil in a large skillet over medium-high heat. When the oil shimmers, add the onion and cook, stirring occasionally, until soft and translucent, about 3-4 minutes.
Transfer the onion to a large mixing bowl. Add the salmon, saltine cracker crumbs, egg, mayonnaise, lemon zest and juice, parsley, and salt and pepper to taste. Stir gently to mix. Try not to break up the chunks of salmon too much; do not over-mix.
Form the salmon mixture into 8 patties, about ½ inch thick.
Place the breadcrumbs in a shallow bowl; season with salt and pepper to taste.
Dredge the salmon patties in the breadcrumbs, coating both sides and pressing gently to adhere.
Refrigerate the patties for at least 1 hour, or until firm.
In the same skillet used to cook the onions, heat butter and oil in a large skillet over medium-high heat until sizzling hot (use enough butter and oil to generously cover the bottom of the skillet).
Place the croquettes in the hot oil, cooking in batches if necessary to avoid overcrowding the pan. Cook until golden brown, about 4-5 minutes per side. Flip the patties only once; reduce the heat to medium if they start browning too quickly.
Drain on paper towels. Serve with seafood sauce, tartar sauce, or ketchup. Garnish with fresh parsley and lemon wedges, if desired.