Preheat the oven to 350. Line a large baking sheet with foil and spray it with olive oil or non-stick spray.
Cut the eggplant in two and place cut side up on the baking sheet. Peel and cut the beet into small wedges. Peel the garlic. Wrap the sliced beet and peeled garlic in foil, spray with olive oil, and place on the baking sheet next to the eggplant.Roast for 30 minutes, or until the beets are fully cooked. Carefully remove from the oven and allow to cool until it's safe to touch.
Use a fork or a spoon to scoop the roasted eggplant insides from the skin and place in a bowl. Discard the skins.
Place the roasted garlic and beets into the food processor. Add the lemon juice, olive oil, tahini, salt, and pepper. Process for several seconds until you have a chunky mixture. Add the eggplant (leave any released eggplant juice in the bowl). Process for a minute or two until you reach a desired consistency - I like mine to be a little chunky. Taste and see if you need more lemon juice, salt, or pepper. Add the chopped parsley (if using), and pulse to combine.
Serve at room temperature or chilled, drizzling with extra olive oil and garnishing with fresh parsley. Serve with crackers, cut veggies, fresh or toasted pita, or use as a spread on sandwiches and wraps.