For the tartar sauce: Mix together the mayonnaise, parsley, capers, mustard, onions, relish and lemon juice in a medium bowl. Set aside.
For the coleslaw: Put the cabbage, carrots, onions and parsley in a large bowl. Mix together the sugar and vinegar in a medium bowl until the sugar is dissolved. Mix in the mayonnaise, mustard, celery salt and black pepper. Pour the dressing into the cabbage mixture and fold. Reserve.
For the oysters: Heat the oil in a heavy-bottomed saucepan until a deep-fry thermometer inserted in the oil registers 350 degrees F.
Mix together the flour, salt and pepper in a shallow bowl. Put the eggs in another shallow bowl and the panko in a third. Dip the oysters first in the seasoned flour, next in the beaten egg and finally in the panko. Slip the oysters into the oil and deep-fry until golden brown; drain on paper towels.
Putting it all together: Spread some tartar sauce on the buns. Place 2 oysters on each bottom bun, and top with lettuce and tomato. Finish with the top buns. Put some coleslaw on each plate and top with 2 more oysters.