In a small bowl, mash the avocado with a fork. Add minced garlic and fresh lime juice. Season with sea salt and pepper, to taste.
Mix until well combined.
Using a peeler or mandolin, slice the cucumber into equally thin strips.
Place about 12 cucumber strips next to each other, on a flat surface, such as a cutting board.
Spread 4 tbsp of mashed avocado evenly over the slices. Then add slices of ham and cheese.
Starting at one end, tightly roll all of the cucumber slices into one long "sushi" roll.
Using a sharp knife, cut the roll into individual servings as shown in the photos.
Secure with a toothpick and place on a serving platter. Repeat with the remaining cucumber strips.
Sprinkle with fresh chopped parsley and serve immediately.