Remove the rounded ends from the cucumber and cut into 1/2 inch thick slices. Scoop out some of the cucumber seed area with a rounded teaspoon, leaving the shells and the others side intact. You are making a little hole in each slice. Chop scooped part and save.
Drain the slices well on paper towels while you lightly beat the eggs together with the chopped cucumber scoopings. Stir in half-and-half, salt, pepper and paprika.
Melt butter in a saucepan and cook the egg mixture gently until set. Stir in the ham and parsley. With a teaspoon, spoon the egg mixture into the holes in the cucumber slices. Arrange on a serving plate.