Pour water or broth into a pot, add diced gizzards, peeled parsnip root and onion plus half a teaspoon of salt if needed. Simmer for about 45 minutes or until the gizzards are tender.
Drain the cucumbers, reserve the juice. Peel and grate them on a coarse grater. In a pan, melt the butter and sauté the cucumbers for a few minutes, stirring occasionally.
Add peeled and diced potatoes, peeled and grated carrot, allspice and bay leaf to the simmering soup. Simmer until the vegetables are tender, about 15 minutes.
Pour in the extra virgin olive oil (if using water instead of broth), add the cucumbers from the pan and about 1/2 cup of juice from the pickled cucumbers. Simmer for about 10 minutes.
Taste the soup and season with salt and pepper if needed. Add the heavy cream, remove from heat and stir in the chives.