Combine mayonnaise, lemon juice, parsley, capers, and garlic. Cover and refrigerate until serving.
Pour salmon in a bowl and flake with a fork. Add the bread crumbs, eggs, olives, green onions, and Old Bay. Mix well and form into 8 patties, each about ½-inch thick, compacting firmly.
Heat oil in a large skillet over medium-high heat. Fry patties in batches (do not overcrowd the pan) for 3-4 minutes per side, or until nicely browned.