In a medium bowl, whisk egg, salt, pepper, and parsley and set aside. Squeeze grated potatoes in a clean dish towel to remove any excess water, then add to egg mixture.

Heat oil in a medium nonstick frying pan over medium heat. Press a spoonful of the potato mixture into your hand, then place a salmon cube into it and cover with another layer of potato to enclose the fish.

Fry for about two minutes on each side, until golden. (Watch closely; they cook quickly!) Drain on a paper towel-lined plate to remove excess oil. Repeat with remaining salmon, adding more oil to the pan if needed.

Assemble the sliders with buns, tartar sauce, salmon, lettuce, cucumbers, and pickled onions immediately before serving.