Cut the potatoes in 1/2-inch dice. Place in a saucepan, cover with cold water, and add 2 teaspoons salt. Bring to a boil and simmer until crisp tender, about 5 minutes; drain.
Prepare the grill for direct cooking over medium heat (350° to 450°F). Preheat the griddle over direct medium heat for 5 minutes.
Prepare the grill for cooking over medium heat (400°F). Preheat the griddle over direct medium heat for 5 minutes.
Make four wells in the hash. Crack an egg into a small bowl. Tip the egg into a well. Repeat with the remaining eggs. Close the lid and cook the eggs until set, about 8 minutes.
Season the hash with additional salt and black pepper. Garnish with parsley. Divide the hash between 4 serving plates. Serve immediately.