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.
I was very clever and bought blueberry Greek Yogurt. That being said, I made this sauce with sour cream. It was very good, but perhaps a bit too much lemon (I used the juice from a rather small lemon, so ~2T. My family thought the sauce would lend itself well to a bit of dill weed as well.