Cut the eggplant into ½-inch thick slices. Line two large sheet trays with paper towels and lay the slices on top in a single layer. Season with the salt all over both sides of the eggplant. Set aside for 30 minutes, then pat each slice dry with a paper towel.
Brush both sides of the eggplant with oil and sprinkle with pepper.
For the grill: Preheat a grill (charcoal or gas) to medium-high heat (400-425°F), with the flames or coals spread evenly under the whole grill. Place the eggplant on the grill over direct heat. Grill until grill marks form and the eggplant is tender, 3 to 4 minutes per side.
Alternatively cook in a grill pan: Heat a large grill pan over medium-high heat. Working in batches, add the eggplant in a single layer. Grill until the eggplant is tender and grill marks form, about 5 minutes per side.
Transfer the eggplant to a serving platter. Sprinkle with parsley. Serve with marinara sauce, if you like.