The parasite is not the disease; the parasite causes the disease.

Similarly, SARS-COV-2 is a virus which causes Coronavirus Disease 2019 (COVID-19) and the Human Immunodeficiency Virus causes AIDS.

People often conflate parasites or viruses with the diseases they cause, and it's practically impossible to eliminate the diseases without eliminating the causative agent, but they are technically distinct concepts.

We use "disease" for maladies caused by infections (virus, bacteria), and we use "disease" for maladies caused by genetics (cancer), caused by chemicals (lung disease), and so forth. So yeah, a parasite certainly would qualify I think.

I guess the same way a virus is a disease, in that a small living thing gets inside you and harms you / causes immune system reactions?

I think technically you get a parasite and then it causes a disease in reaction, but if it's a parasite you can spread it's basically fine to model it / talk about eradicating it in the same way right?

How would you draw the line? The common cold, tuberculosis, malaria, and hookworm are all caused by foreign bodies which enter the human body, reproduce, and cause illness; the only difference is the size of those agents (virus, bacterium, single-cell parasite, multicellular parasite).