Is it possible for liver parasites to cause gallstones?

Yes. It is not common in the US bu in Asia you can get parasitic infection with a liver fluke called Clonorchis sinensis, and get a gallstone as a result.
Actually yes. Some parasites in asia i think cause debris in your biliary system which acts as a nidus for stones. But these are uncommon parasites and you need to consult a doctor if you think you have one.