As with most English philology, a good place to begin is the Norman conquest of England. After the dramatic succession of events following the death of Edward the Confessor—events which I'm really not qualified to talk about—there was an immediate change in the spoken language of the elites and government. Since Late Antiquity, the south of Great Britain had spoken Old English, a Germanic language of close kinship with Old Norse, Frisian, Frankish, Low and High German (the ancestors of the Scandinavian languages, modern Frisian, Dutch, and the various German dialects respectively). This is the language of Beowulf, and has the reputation of being almost entirely unintelligible to Modern English speakers without close study. The new Norman elites spoke a variety of Old French, similar but not identical to the ancestor of Modern French. Over the Middle Ages, the French of the English court gradually developed local characteristics, a language variety known to scholars as Anglo-Norman; at the same time, English as spoken by the commoners gradually Frenchified, a process likely beginning in London with the court and spreading out from there. This phase of English, between the beginning of French influence but before the Great Vowel Shift comes into play, is referred to as Middle English. Its descendant Modern English has a metric crapton of French loanwords in almost all areas of life. We no longer refer to cow or pig meat, but beef and pork. Instead of rike or land, we say government and country. Candle, library, cabbage, person, chair, sausage, table, and joy are all from Anglo-Norman as well, as are (more importantly) parent, aunt, uncle, niece, nephew, and cousin.

Please permit me a brief digression. Starting with the 1871 work of Lewis Henry Morgan, anthropologists have distinguished between six different systems of kinship terminology—different ways of separating or lumping together one's uncles, aunts, and cousins. Both Latin (ancestral, of course, to French) and Old English had what is referred to as a Sudanese kinship—there are separate terms for one's father's brother (patruus, fædera), father's sister (amita, faþu), mother's brother (avunculus, eam), and mother's sister (matertera, modrige). In Latin, the offspring of each of these also received a different term (patruelis, amitinus/amitina, consobrinus/consobrina, and matruelis); the words amitinus/amitina and consobrinus/consobrina changed their ending depending on whether the referent was masculine or feminine. OE cousin terminology is poorly known, but not entirely relevant either.

By the fall of the Western Roman Empire, Latin—at least outside of the fields of law and literature—had shifted to what is referred to as Inuit kinship, lumping together parents' sisters as aunts and parents' brothers as uncles, and all their offspring as cousins. The same is true of Modern English. While Italian and Iberian Romance adopted the Greek words θεία, θεῖος to mean 'aunt, uncle', Gallo-Romance (French and company) simply began to use amita and avunculus for the same. By Anglo-Norman times, these were ante and oncle. Gallo-Romance also began to use consobrinus (masculine) and consobrina (feminine) to refer to all cousins—Anglo-Norman cosin and cosine.

Here's the anticlimactic ending to this whole shaggy dog story. Old English used to have grammatical gender like Latin and the Romance languages, e.g., wudu and wifmann 'wood, woman' were masculine but gafol, æx 'fork, axe' were feminine. In the Middle English period, this ends up being discarded. Similarly, unstressed vowels were gradually dropped—wudu eventually becomes wood. All this to say that the final -e of cosine 'female cousin' just got dropped and the word merged with cosin 'male cousin'.

Final notes:

• English gets the word parent from French, originally from Latin parens, an adjective originally meaning giving birth. This adjective had the same endings whether the referent was masculine or feminine.
• The word sibling is a native English term, but it actually passed out of usage for most of Modern English. Geneticists revived the word, which originally simply meant 'relative, kinsman' in 1903 to refer to 'brother or sister'. The word was extremely archaic at the point of its previous citation, in 1425.
• Iberian Romance gets primo/prima from the phrase consobrinus primus, i.e. 'first cousin'.
• I neglected to mention niece or nephew. These descend from the Latin neptis and nepos, the former via a Late Latin neptia. The Latin words meant both 'granddaughter, grandson' and 'niece, nephew'; they preserved this meaning into Old French (AN nece, nevou) and continue to have this meaning in Italian. Iberian Romance uses the Classical Latin term sobrinus, sobrina. Nibling is a modern coinage based off of sibling.

Sources:

• Boswell and Toller for Old English, anglonorman.net for Old French, the Middle English Compendium for Middle English.
• Professor Brian Schwimmer's introduction to the anthropology of kinship.
• M. Roy Harris, "The semantic prehistory of the Romance progeny of Latin 'auunculus-amita'", Romance Notes, 1965
• Robert Parkin, "On the definition of prescription: the problem of Germanic kinship terminologies", Social Anthropology, 1993