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 3 3/4 cups 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.
Pour into a serving bowl and stir in yogurt. Chop the remaining 1/4 cup cucumber slices.
Refrigerate and serve it chilled.
Just before serving, garnish with the chopped cucumber and more chopped parsley,
Refrigerate for up to 4 hours.