If using small eggplants: Trim away the green tops and slice them in half lengthwise. Make small incisions in the flesh in a crisscross pattern, taking care not to cut so deeply as to puncture the skin.
If using large eggplants: Trim away the green tops and slice across their width into disks 1½inches thick. On one side of each disk make crisscross incisions spaced 1 inch apart, taking care not to cut through to the other side.
Spread the eggplant slices on the inside wall of a large colander set over a bowl and sprinkle liberally with salt. Let stand for about 30 to 45 minutes, allowing as much as possible of their bitter juice to drain away.
Turn on the oven to 450°.
Choose a baking pan large enough to accommodate all the eggplant slices without overlapping. Smear the bottom lightly with a little bit of the olive oil. Place the eggplant in the pan; if using halved small eggplants, lay them skin side down; the slices of large eggplant, crosshatched side up.
Sprinkle the garlic and parsley over the eggplant, distributing it as uniformly as possible and pressing it into the cuts. Add a pinch of salt and several grindings of pepper. Pour half the olive oil over the eggplant, guiding it as much as possible into the cuts.
Place the pan in the uppermost level of the preheated oven. After 15 minutes, remove the pan, pour the rest of the olive oil over the eggplant, and return it to the oven. Bake for 15 or more minutes, until the eggplant feels as tender as dense cream when tested with a fork. Serve hot, lukewarm, or at room temperature. Do not refrigerate and reheat.