Begin by making the jerk marinade. Combine all the marinade ingredients in a blender and process until you have a smooth puree.
Add chicken pieces into a large resealable plastic bag and pour in the marinade. Put the bag into a baking dish and let marinate in the refrigerator overnight.
Preheat grill to high.
Prepare a table-top smoker or take a roasting tray, line it with foil and add some soaked wood chips to the bottom. Place a wire rack over the top, upside down, and lay chicken pieces over the chips on the rack. Cover with foil and grill over high heat. Smoke for 10 minutes.
Preheat oven to 300 degrees F.
Transfer the chicken pieces to a baking sheet and drizzle with extra-virgin olive oil. Bake in the oven "low and slow" until chicken is tender, about 1 hour. Remove the chicken from the baking sheet to a serving platter and garnish with limes and parsley before serving.