Parashurama is also sometimes referred to in translation as "Rama-(or Ram)-with-the-axe", or "axe-wielding Rama", or order to distinguish him from the Lord Ram described in the Ramayana. The story of his incarnation is that the warrior-king caste, the Kshatriyas, had rebelled against the Brahman caste, and were asserting themselves to be the superior caste.
Such an ordering of affairs went against the natural order of things, and violated the dharma of the different castes- which the legend maintains insisted on the superiority of the Brahman order. Vishnu was moved to once again take up his role as restorer of the law and proper order, and incarnated as the Brahman Parashurama. Parashurama went out among the Kshatriya kings, and with his axe killed all those who asserted the superiority of their caste, and refused to acknowledge their proper place in the hierarchy of the caste system. The notion of a Brahman priest going about walloping folks with an axe seems somewhat counter to convention, but as the tales associated with Krishna demonstrate, when the gods walk the earth, the regular standards of conduct and propriety can seem reversed. This is sometimes explained by pointing out that incarnation, like embodiment in a religious image or icon, is no more than lila, the play of the gods. They give the appearance of certain actions or aspects, but the ultimate truth is always other than that which humans perceive.
Whatever the theology surrounding this incarnation of Vishnu, historically it offers some interesting insights. Counter to the conventional wisdom, the Brahmans have not always been the highest of the castes in the Indian subcontinent. In ages past that role was filled, in both civil and religious functions, by the Kshatriya caste. We known from preserved documents and discourses that at the time of the Buddha (around the 4th or 5th Century BC), a controversy arose as to which caste would become the highest throughout the reaches of Indian culture.
Some of these controversies, and the associated debates, are recorded in the Buddhist Pali Canon- the Digha Nikaya, in particular, contains a recording of an interaction between the Buddha and a Brahman who refuses to pay his respects to the Buddha, because he feels that such niceties are not due a Kshatriya. The Buddha both asserts that the Kshatriyas are the conventional higher caste in Indian society, and at the same time maintains that moral behavior is the proper determination of an individuals purity and significance, rather than caste.
Whatever the practice at the time of the Buddha, it's clear from history that by the time of extensive Western contact with India (indeed, probably by the time of the Mogul conquest of the subcontinent), the Brahman caste had emerged as the victor from the caste controversy. The legend of Parashurama may represent an attempt by the Brahman caste to give their ascension to the highest order a historical and religious context- representing the superiority of the Brahman caste as the religious and historical norm, and the primacy of the Kshatriyas as a temporary deviation from the proper dharma. While the legend and texts where it is mentioned most likely predate the actual debate, it seems possible that this was an issue that may have emerged multiple times in history. Furthermore, the versions, commentaries, and references we have now have likely been influenced by the 4th Century debate, and its ultimate resolution.
And for those who prefer their religious texts uncolored by historical inquiry, given the incomprehensible time scale on which Hindu and Buddhist myths occur, it is entirely possible that the tale of Parashurama records a litteral truth- but who knows in which world or age it truly occured!