Gently pick over the crab meat and remove any bits of shell.
Crack the eggs into a large mixing bowl and beat with a fork.
Add the mayonnaise, mustard, Worcestershire sauce, lemon juice, Old Bay, and salt. Stir to combine thoroughly.
Add the crab meat, panko, and parsley to the bowl.
With a large spoon or a silicone spatula, gently fold the ingredients together to combine well without breaking up the crab meat.
Line a baking sheet with parchment — you'll use it to chill the crab cakes after forming and before cooking.
Divide mixture into 12 equal parts using a dry ½-cup measure.
Gently form each portion into a relatively tidy mound and place onto the lined baking sheet.
Cover tightly and refrigerate for at least an hour, up to 24 hours.
When ready to cook, pour safflower oil into a 12-inch nonstick frying pan to a depth of about ¼ inch. Warm over medium-high heat to between 350° and 375°F.
Add half the crab cakes, being careful for splatters. Cook undisturbed until golden brown on the undersides, likely four to five minutes. It is very important that the crab cakes have formed a nice, cohesive crust on the undersides before you flip them, or else you'll have trouble getting them to hold together.
Just before flipping, give each cake a gentle pat with the spatula to compress the ingredients just slightly. This will minimize any loose pieces escaping the cakes as you flip.
Carefully flip each cake. I sometimes like to do this with two spatulas, as you'll see in the video. Continue cooking until golden brown on the second side and cooked through, another four minutes or so.
Remove from pan and let rest on paper towels or a wire rack. (If you like, you can keep them warm in the center of a 200°F oven.)
Repeat with remaining crab cakes, adding more oil to the pan if necessary, and serve.