Oil the grates of a grill and set it to medium-high, or oil a grill pan and place it over medium heat on the stovetop.
Toss asparagus in 1 tablespoon olive oil, 1/4 teaspoon salt, and 1/4 teaspoon pepper.
Grill asparagus for five minutes per side, until grill marks are formed and asparagus is tender but still bright green and pliable, but not limp.
Toss lemon slices in remaining olive oil. Grill over high heat for about two minutes per side, until the lemons are blackened in spots. Note that lemons should be well cooked because doing so removes most of the bitterness that would otherwise be in the pith.
Remove lemons from grill and chop each entire slice finely ( it will be mushy), including the peel. Discard any seeds.
Place the asparagus in a bowl and toss with the chopped lemon, and remaining salt and pepper. Sprinkle with parsley, if desired, to serve.
Notes
Note that this Grilled Asparagus and Lemon can be made into an entree, by chopping the asparagus into bite-sized pieces after grilling, tossing with the lemon, and adding in flaked wild salmon (either from a cooked fillet or a can).