In a large bowl, combine mayonnaise, beaten egg, Dijon mustard, Worcestershire sauce, Old Bay seasoning, salt, and black pepper. Mix well until smooth.
Gently fold in the lump crab meat, crushed saltine crackers, chopped parsley, and chopped green onions. Be careful not to break up the crab meat too much.
Shape the mixture into 8 equal-sized patties, about 3 inches in diameter and 1 inch thick. Place the patties on a baking sheet and refrigerate for at least 15 minutes to help them set.
Heat the butter and vegetable oil together in a large skillet over medium heat until the butter is melted and the mixture is hot.
Carefully place the crab cakes in the skillet, cooking in batches if necessary to avoid overcrowding.
Cook the crab cakes for 4 to 5 minutes on each side, or until they are golden brown and crispy on the outside and heated through.
Remove the crab cakes from the skillet and place them on a paper towel-lined plate to drain any excess oil.
Serve the crab cakes hot with lemon wedges and your favorite dipping sauce.