Preheat the grill.
Quarter the avocados and remove the seed. Season with salt and pepper, to taste, and brush with grapeseed oil. Grill the avocado slices over indirect heat, leaving undisturbed for the first 2 to 4 minutes or so to let the seasonings integrate into the surface and to allow the caramelization process to begin. Then, flip the avocados, remove from grill and let rest.
Through the feed opening of a running blender add, 1 at a time, the vinegar, salt, black pepper, mustard, parsley, and garlic. Leaving the blender running add the olive oil through the feed opening in a slow thin stream.
Serve the grilled avocado with enough of the dressing to coat and garnish with fresh parsley sprigs.