Why is a mosquito feeding on human blood not a parasite?
A mosquito is a biological parasite, it is not a medical parasite. 

There are two definitions of parasite. A biological/ecological definition and a medical/physiological interaction definition.


A parasite in biological terms is an organism that benefits from a parasitic relationship; a parasitic relationship being a non-mutual relationship between species, in which one species benefits at the expense of the other. Generally the host is not killed by a small number of parasites. When the host is killed the organism is usually called predator or parasitoid. 
A parasite in medical terms is an organism that lives on or in a host and gets food from or at the expense of its host. 


The difference is small but important; only the medical definition requires the parasite to live in or on the host for prolonged periods. It is a much narrower definition.

Biologically, a female mosquito is an indirect ectoparasite, it can be facultative or obligate depending on the species. It harms its host to benefit itself, that is all that is needed to be a parasite by the biological/ecological definition. And just like a leech or vampire bat it is hemophagic and leaves the host as soon as it is done feeding. Brood parasites are another great example of a biological parasite that does not live on or in the host. So a cuckoo would be a biological parasite but not a medical parasite. When you consider the function and practice of medical science the more narrow definition makes sense, they are not concerned with parasites that are not going to stick around or not affect the host organism's physiology directly. By the narrower medical definition, none of these organisms are parasites even though by the biological/ecological definition they are. 

Consider the wiki or a paper on parasite evolution vs say the CDC to see the difference.