Mix together the tuna, mayonnaise, lime juice, onion, and parsley. Place in the refrigerator.
Boil the potatoes until soft; about 10 minutes. Drain and let cool.
Put the potatoes through a ricer.
Mix in the rest of the potato layer ingredients.
Line the inside of a loaf pan with plastic wrap. Coat with nonstick cooking spray.
Spread a layer of the potatoes on the bottom; about a third of the way up.
Place the sliced avocado across the potato layer.
Pour the tuna salad over the avocado layer.
Spread the rest of the potatoes over the tuna salad layer.
Tighly wrap the loaf pan in plastic wrap. Place in the refrigerator for 4 hours before serving to firm up.
Once firm, flip onto a plate.