I recently read in my Ecology course notes that a mosquito feeding on human blood is not considered as a parasite. However, since it sucks blood from the human body, shouldn't it be regarded as a parasite, just like lice and ticks?
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.
Female* mosquitoes are generally not to be considered ectoparasites because they spend so little time with the host. Instead, they are sometimes classified as micro-predators.
*Male mozzies don't practice hematophagy at all
According to some sources, the term parasite hinges on the time the parasite spends on, or in its host and hence it is a gray area whether a mosquito is considered a parasite or not.
The CDC, a renowned center on disease control & prevention, states the following:
Although the term ectoparasites can broadly include blood-sucking arthropods such as [female] mosquitoes (because they are dependent on a blood meal from a human host for their survival), this term is generally used more narrowly to refer to organisms such as ticks, fleas, lice, and mites that attach or burrow into the skin and remain there for relatively long periods of time (e.g., weeks to months).
In contrast, female mosquitoes need only about 4 minutes to finish their drink.
Females of most species are ectoparasites...
But then, when clicking through to the Ectoparasite Wiki page no mention of mozzies are made and the more accepted parasitic arthropods are discussed, namely the ones stated above by the CDC (lice, ticks and fleas).
However, as mentioned in the comments below, with credits to Fileunderwater, other sources like this veterinary work on Ectoparasites also discusses mosquitoes.
Hence it seems to be a gray area whether mozzies are classified as (ecto)parasites. Personally, I would stick to the careful definitions provided by credible sources like the CDC - mozzies are not considered ectoparasites, because they spend just a few minutes siphoning blood before they are off again.
But then, as rightfully probed in the comments, what are they then if not parasites?
One classification I've found for insects practicing hematophagy for short periods at a time is
Micro-predator: a temporary parasite (source: English Encyclopedia);
Micro-predator: e.g., the mosquito, that derives elements essential for its existence from other species of organisms, larger than itself, without causing their destruction (source: Medical Dictionary).
To be a parasite, an organism needs to fulfill two criteria
1) It needs to have a non-mutual relationship between itself and its host. The organism derives benefit and the host suffers harm.
2)The organism also needs to live in or on their host for an extended period.
Often, a parasite will also have biological specialization to mediate the relationship between itself and its host.
However in reality, there is a spectrum of relationship. So a parasitism is not always very clear cut. How much "harm" is needed to be harmful enough? How long does a relationship need to be?
In your example the first criterion is fulfilled, but the second is not. The mosquito benefits from the blood meal and the host mammal does not. However, only female mosquitoes feed on blood and only when they are about to lay eggs. A mosquito does not spend most of its life in or around its host.
So by definition a mosquito is not a parasite.