Blend 3 pints strawberries, 1/2 the cucumber, 1/2 the onion, 1/4 cup cilantro, and 1/4 cup parsley in a blender on high speed until pureed, about 30 seconds. Pour the pureed mixture into a large bowl.
Stir 1 pint strawberries, 1/2 the cucumber, 1/2 the onion, 1/4 cup cilantro, 1/4 cup parsley, green onions, jalapeno pepper, red wine vinegar, lemon juice, olive oil, salt, garlic, tarragon, basil, hot pepper sauce, and black pepper into the pureed strawberry mixture. Spread avocado cubes over the top of the gazpacho.
Cover the bowl and chill gazpacho thoroughly, at least 2 hours, before serving.