Heat grill to medium-high. Halve eggplants lengthwise; slice off enough from skin side so halves lay flat. Brush both sides of eggplant halves with oil; season generously with salt and pepper. Grill until tender, 5 to 7 minutes per side.
Meanwhile, in a small bowl, combine cucumber, yogurt, parsley, and lemon juice; season with salt and pepper. Serve sauce with grilled eggplant and lemon wedges, if desired.