With a sharp knife, cut dates from pits lengthwise into 1/4-inch-wide strips; discard pits.
In a bowl, gently mix the celery with salt to taste (the mixture should taste slightly salty).
Mound celery equally on salad plates. Scatter dates, separating pieces, over celery; top equally with parmesan slices. Drizzle about 2 teaspoons vinegar over each salad.