Pre-heat your grill to medium-high heat.
Cut off the green stems and the very bottom of the root (but not so much that the layers have nothing to attach to). Then cut the fennel in half lengthwise, and then again into 4-6 bite-sized wedges.
For an outdoor grill, simply brush your fennel wedges with olive oil, sprinkle with sea salt and grill until soft and tender, turning occasionally.
For a grill pan, heat the pan on medium high heat for 3-5 minutes. Lightly coat fennel in olive oil and sea salt (use a bowl and stir). When the pan is hot, add 1-2 tbsp olive oil and gently swirl it in the pan so it coats the surface. Place fennel in a single layer on the hot grill, lower the heat to medium and cook until translucent, tender and slightly browned, turning occasionally, approximately 10 minutes.
Your fennel should have grill marks and be caramelized in places. Be patient: and allow the fennel to become tender. Of course, you can choose your desired crunchiness. Remove the fastest cooking fennel pieces from the grill when they are done and place them in a bowl.
When all the fennel is finished cooking, drizzle it lightly with lemon oil (or juice and zest) and sprinkle with freshly chopped parsley. Adjust salt and zest if necessary.