I believe that parasites can only survive in a body that is not in balance, is toxic and malnourished. In the same way Gallstones are only produced in a body that is not functioning optimally and is not in balance. Water Fasting provides the environment in which the body can heal itself and attain that balance, so perhaps indirectly it assists with these problems.
All the parasite cleanses and Liver Flushes in the world will not help you unless you change your eating and living habits. I also believe that some of these 'remedies' (including water fasting) can be very depleting to the body and unless they are done responsibly can do more harm than good in the long term.