Scrape skin of cucumbers lengthwise with fork to make long grooves. Cut in 2-inch sections and cut out seeds, forming rings.
Mash anchovies with dill, chives, cream cheese, and sour cream. Season with salt and pepper. Fill cucumber rings with this mixture and chill in refrigerator for 2 hours.
Garnish each cucumber with a teaspoon of sour cream, then a teaspoon of caviar. Serve with lemon and parsley.