With a heavy sharp knife, chop tuna into very small slices. In a mixing bowl, combine tuna, chives, capers, shallots, lemon juice, oil, salt and freshly ground pepper and chill for 1 hour.
Whilst tuna is chilling, prepare cucumber boats. Peel cucumber and cut into 2-inch long cylinders. With a teaspoon, scoop out the seeds from inside the cucumber disc without touching the very bottom of the cucumber. (In other words, leave a "bottom" in the hollowed-out center of the cucumber.)
Season cucumber with salt and pepper. Fill with 1-ounce of tuna tartare. Top with 1/8 teaspoon of caviar. Finish with a sprig of parsley.