In a large pot, heat the oil over medium heat. Add the onion; cook until slightly softened. Add the potatoes, chicken broth and 1 1/2 cups water. Bring to a simmer, cover and cook until the potatoes are tender, about 15 minutes. Using a food processor, and working in batches, puree the potato mixture with the cucumber. Return to the pot and stir in the cream.
Heat a heavy skillet over medium-high heat. Add the corn and cook until golden, about 5 minutes. Scrape off the corn kernels; add to the soup. Stir in the parsley; season with salt and pepper. For cold soup, refrigerate until chilled.