Heat oil in a large saucepan over medium-high heat. Add garlic and onion; cook, stirring occasionally, until tender, 1 to 4 minutes. Add lemon juice and cook for 1 minute. Add almost all of the cucumber slices, broth, salt, pepper and cayenne; bring to a simmer. Reduce heat and cook at a gentle simmer until the cucumbers are soft, 6 to 8 minutes.
Transfer the soup to a blender. Add avocado and parsley; blend on low speed until smooth. (Use caution when pureeing hot liquids.) Pour into a serving bowl and stir in yogurt.
Chop the remaining 1/4 cup cucumber slices. Serve the soup warm or refrigerate and serve it chilled. Just before serving, garnish with the chopped cucumber and more chopped parsley, if desired.