A pathogen is defined as anything that is able to cause a disease within a host. As such, a pathogen can be one of many types of microorganisms or particles such as bacteria, fungi, viruses, prions and parasites. This broader definition also means that the term pathogen is not restricted to something that causes a disease in human. It can also cause a disease in any living organism such as an animal, a plant or even another microorganism. Furthermore, as pathogen can be quite specific for its definite host, a microorganism or a particle that is pathogenic for one type of hosts for example a specific species of plants is not necessarily pathogenic for other organisms, such as animals or humans in that case. As such, we can say that all parasites causing a disease in human are pathogens, but it is incorrect to say that all pathogens are parasites, as they can be other types of microorganisms, viruses or prions. Following this definition, it is also incorrect to say that all parasites are pathogens as some are able to live as free-living organisms.
Following this broader definition, all parasites that cause a disease in a specific organism are considered as pathogens for this specific organism. It can also be said that all parasites causing a disease in human are known as pathogens for human. However, the vast majority of the known parasites are not pathogenic for humans, although they can be pathogenic for other organisms. For instance, a scientific study suggests that there are approximately 36,400 species of protozoan parasites on Earth and within the ocean. However, it is very difficult to find a prediction of the total number of helminth species present in the world as they are usually classified with the animals. Among them, around 70 species of protozoan parasites and around 300 species of helminths are known to have human as their host and are then considered as pathogens for humans. However, many of them are extremely rare or have human only as accidental host, leaving us with approximately 90 common parasite species pathogenic for human.
As such, parasites are found everywhere in the environment and can even be beneficial for their specific hosts. For example, some protozoan parasites species are responsible for the digestion of the cellulose present in grass within the rumen of cow and present in wood within the guts of termite. In these cases, these parasites cannot be considered as pathogenic for these species, as they are not causing a disease.
Finally, parasites can or cannot be considered as pathogens depending on the host species involved. Furthermore, other microorganisms and particles can also be known as pathogens, a term not restricted to parasites.
Sources: Cox, F. E. (2002). History of human parasitology. Clinical microbiology reviews, 15(4), 595-612; Mora, C. et al. (2011). How many species are there on Earth and in the ocean?, PLoS Biol., doi: 10.1371/journal.pbio.1001127.