Cut the eggplant into halves horizontally that cut each half vertically into two. Place in a bowl. Pour in the olive oil. Toss to coat every square inch of the eggplant slices.
Heat a non-stick pan. Make sure it's really hot before adding the eggplant slices. If the pan is not hot enough, the eggplants will steam rather than brown and that will make them soggy.
When the pan is hot, place the eggplants side by side, cut side touching the bottom of the pan. Let them cook for a minute or two without disturbing them. This isn't stir frying so you need to overcome the urge to move them around. If you do, they won't brown properly.
Flip the eggplant slices over. Sprinkle with celery salt and pepper. Lower the heat to medium and cover the pan. Cook for about two minutes.
Flip the eggplant slices again and cook for a minute. Flip them over one last time and sprinkle the cut sides with grated Parmesan. Cover the pan and turn off the heat. Let the eggplant slices finish cooking in the residual heat. The cheese will also soften during this stage.
Finally, sprinkle the eggplant slices with parsley.
Serve at once.