While all pathogens act parasitically, a definitional distinction is made between Eukaryotic parasites; protists, worms, and ectoparasites such as lice, and pathogens such as bacteria, viruses, and even eukaryotic fungi.
The Center For Disease Control's resource on Parasites states:
A parasite is an organism that lives on or in a host organism and gets its food from or at the expense of its host. There are three main classes of parasites that can cause disease in humans: protozoa, helminths, and ectoparasites.
The World Health Organization fact sheet on Antimicrobial Resistance refers to parasites separately from Bacteria, Viruses, or Fungi:
Antimicrobial resistance threatens the effective prevention and treatment of an ever-increasing range of infections caused by bacteria, parasites, viruses and fungi.
The American College of Physicians makes the same distinction on its page describing the specialty of Infectious Diseases:
ID specialists have expertise in infections of the sinuses, heart, brain, lungs, urinary tract, bowel, bones and pelvic organs. Their extensive training focuses on all kinds of infections, including those caused by bacteria, viruses, fungi and parasites. Many ID physicians specialize in treating patients with infections due to human immunodeficiency virus (HIV), the cause of AIDS.
Even the Wikipedia page on Parasitic Diseases points out the distinction:
Although organisms such as bacteria function as parasites, the usage of the term "parasitic disease" is usually more restricted. The three main types of organisms causing these conditions are protozoa (causing protozoan infection), helminths (helminthiasis), and ectoparasites.(1) Protozoa and helminths are usually endoparasites (usually living inside the body of the host), while ectoparasites usually live on the surface of the host. Occasionally the definition of "parasitic disease" is restricted to diseases due to endoparasites.(2)
In my opinion this is an important distinction to make. The treatment protocol for prokaryotic organisms and eukaryotic organisms are different. Evolutionarily, eukaryotic organisms are far closer to us than prokaryotes, and so strategies such as disrupting the ability of a ribosome to translate proteins, a method that some classes of antibiotics employ, would not only affect the protist or worm, it would more than likely affect our own cells ability to produce proteins. In bacteria, the ribosomal subunit is different enough that it can be targeted by medicines, yet similar enough that we can use them in recombinant DNA technology to reproduce mammalian genes. The closer we get evolutionarily to the pathogen, the more alike their molecular machinery is to our own.
Even our own immune system makes a distinction. Leukocytes such as eosinophils are usually only raised to combat parasitic infections. Organisms that are medically termed as parasites also pose a problem in that they are often larger, protists, or much larger, worms, and our phagocytic cells such as macrophages and neutrophils are unable to phagocytose them as a way of clearing an infection.
From your question, you ask
If parasites are organisms that feed on other living organisms, then aren't all pathogens parasites?
I've emphasised on 'feed' because I guess the pathogens just eat us (i.e. take nutrients to reproduce and sustain their lives), right?
There are a host of organisms that live in and on the human body, that feed and reproduce in us and on us that do not do us any harm, known as commensals, and might even be symbiotic, gut flora. However commensals and even symbionts can do harm if they gain access to areas of the body other than their niche locales.
In fact, immunologists often view commensal organisms as part of the bodies innate immune defenses against pathogens. Because they take up space and use resources that would otherwise be used by pathogenic organisms, then serve to limit the ability of pathogens to reproduce and to gain access to tissue that would be vulnerable to the pathogen.
Commensal organisms can become opportunistically pathogenic if they gain access to sterile environments within the body that have not evolved to live "in harmony" with these organisms. An example of this is peritonitis. We can live our lives perfectly fine when our intestinal bacteria remain within the intestines and the colon, however if those tissue are damaged and perforated, these organisms that posed no risk to our health, and actually probably provide a benefit can lead to serious infection and possible death.
There are also commensal Eukaryotes, such as mites and fungi that either do no harm or are kept in check by a properly functioning immune system.
You also ask:
But then there's another question.
If, pathogens need us (a host) to survive, then why do they eventually kill us? I mean, why did they evolve to become an organism that exploits their food to the extent that even they, themselves die? Isn't it disadvantageous for pathogens to kill their hosts, because they will eventually die?
Or I'm totally wrong, do those pathogen sustain their lives even when host has died?
Or perhaps they don't care if they, as individual organisms die, because they can survive as an species by infecting other.
I think where you might be going wrong here is that you are thinking in terms of sanitary and clean forms of death and burial that is a social construct of human primates. Through much of the evolutionary history of multicellular organisms, when organisms died, they died in place and decomposed. This decomposition allowed the pathogenic organisms a route to go on and spread to other organisms. Other living organism came in contact with the decomposition and the pathogen had a way of spreading.
Also many bacteria and viruses can survive outside of their host for extended periods of time, so they in effect are dormant in the environment until they come in contact with a new host that they have the opportunity to infect. In this case, the most important thing for them is to escape the host until another suitable host comes along that can sustain an infection.
Though there are pathogens that have evolved the ability to circumvent our social norms. The body of a victim of the Ebola virus is highly contagious and can infect those that ritualistically tend to the body for burial. The WHO has developed guidelines to try and respect the cultural customs of burial while also preventing the spread of the disease. You could also think of Cholera. Vibrio cholera reproduces extremely rapidly in the host, and the toxins that it produces cause the host to expel the pathogen at a rate that can kill. This serves cholera as it can then go on to infect a new host.
You also have to remember that there are pathogens that are pathogenic in one organism but commensal in others. The parasite that causes malaria is commensal in mosquitos. The vector for Ebola is thought to be bats, but bats do not succumb to the disease. So for pathogens like this, the strategy would be to find a reservoir vector, then find a host where they can replicate, then escape that host either to reinfect another host or find another carrier where it can survive commensally.