There is no universal answer for this question as parasites are part of a highly heterogeneous group. Even though it is not a strict rule and exceptions are always possible, intestinal helminths and some protozoan parasites tend to make you loose weight while there is some new evidences suggesting that other types of intestinal protozoan parasites might make you gain weight instead.
Actually, weight loss combined with an increase of appetite is usually one of the typical symptoms related to infections with intestinal parasites. This type of situation occurs because the infected person must feed its internal parasites. As such, a part of the food and nutrients ingested by the human host is taken away by the parasites. This is the reason why the person is loosing weight even though experiencing an increase of its appetite. Furthermore, in some cases where gastrointestinal symptoms are severe, weight loss can also be a consequence of these symptoms rather than a symptom of its own. In these cases, the severe infection is usually not accompanied by an increase in appetite, as the host is very ill.
As weight loss is a common symptom of infection with intestinal helminths, it is interesting to note that some dieters have even tried a new type of diet based on tapeworm infection in order to loose weight. Even though tapeworm pills for dieters are illegal in most countries, they seem to be available on the black market and on the Internet. People usually think that tapeworm infection is harmless because it is usually asymptomatic, but this type of diet can be very risky and dangerous as potential side effects caused by the tapeworm infection can occur, including mild to severe intestinal illness and symptoms of malnutrition. Dissemination of cysts within the body, a condition called cysticercosis, is also a rare complication of the ingestion of Taenia that is usually more complicated to treat and may require surgical removal of the cysts. Bowel obstruction caused by too many worms is also a severe complication of tapeworm infection requiring usually urgent surgical procedure. As such, this tapeworm diet is risky and not a very good idea in order to loose some weight without doing any effort.
On the other hand, preliminary research suggests that some intestinal protozoan parasites might be somehow linked to weight gain, but further study will be required to validate this hypothesis. Actually, some researchers found that a supposedly harmless intestinal parasite infecting an insect, namely the dragonfly, causes some metabolic disorders similar to those observed in human suffering from obesity. Further investigations will be required in order to validate the presence or not of this parasite in the human gut and to link it with human obesity. Moreover, the interaction of parasites and bacteria in the human gut will have to be assessed, as it has already been suggested that obesity might be linked to the fluctuations of the species of microorganisms present in the gut. However, it seems that a partial clue comes from AIDS patients that are chronically infected by Cryptosporidium, a protozoan parasite related to the one used in the dragonfly experiments, and exhibiting the same metabolic disorders than the infected dragonflies. Finally, this study also showed that, even though some parasites are considered as harmless, they are therefore susceptible to have some effects within the body just by modifying their environment.
Sources: Centers for Disease Control and Prevention
Schilder, R. J., & Marden, J. H. (2006). Metabolic syndrome and obesity in an insect. Proceedings of the National Academy of Sciences, 103(49), 18805-18809.