Pour oil into a small skillet, and turn the heat up to medium. Add garlic when oil is warm and cook, stirring until fragrant, being careful not to burn the garlic. Turn off heat and set pan aside.
In a blender, pour in the garlic and olive oil then add peeled and pitted avocados, vegetable broth, lemon juice, salt and pepper. Blend together. Taste and season as appropriate.
Pour into a container and place in refrigerator to cool.
Meanwhile, in a small bowl combine crab, tomato and cucumber. Season with salt and pepper. When soup is chilled, pour into bowls and top with crab cucumber mixture. Sprinkle with chopped parsley and enjoy!