Bring a large pot of water to a boil and gently add the eggs so as not to crack them. Return the water to a boil, then reduce the heat to keep them at a gentle boil. Cook the eggs for 13 minutes, then drain them and put them in cold water. Change the cold water three times.

Place the tuna in a mortar and add 1 tablespoon of the mayonnaise. Using the pestle, work the tuna and mayonnaise together until a smooth paste is formed; add more mayonnaise if needed. Add this paste to a medium bowl, add the remaining mayonnaise and the tuna oil, and combine well to blend to a uniform paste with no lumps.

Cut the eggs in half lengthwise and gently remove the yolks. Reserve one of the cooked yolks for garnish. Place the remaining yolks in the bowl with the mayonnaise paste. Mix the egg yolks into the paste until smooth. Season with lemon juice and lightly salt to taste (the tuna and its oil will be fairly salty already).

Arrange the 16 egg white halves on a serving dish. Spoon the yolk mixture into a pastry bag fitted with the star tip. Pipe the filling into the white halves. Alternatively, spoon the mixture into the halves.

When all the egg white halves are filled, rub the reserved egg yolk through a sieve in a shower over the filled eggs. Garnish the plate with parsley.