TL;DR: The text does not say.
The feature of the text that most strongly suggests that my true love might have sent twelve partridges is the word ‘On’ at the start of each verse. But this word is an interpolation, and does not appear in the earliest printed version:
Anonymous (c. 1780). Mirth Without Mischief, p. 16. London: C. Sheppard.
The question asks us to treat the song as if it were documentary evidence for a fictional universe in which someone’s true love sends them a number of presents during the twelve days of Christmas. But turning the question around, what do we gain by treating the song in this way? The song does not seem to be describing a fictional situation in a documentary way, and to insist on that approach just runs into the limitations of the text.
A look at the origin of the song gives us some insight into why the “documentary evidence for a fictional universe” approach doesn’t seem very productive in this case.
“The Twelve Days” was a Christmas game. It was a customary thing in a friend’s house to play “The Twelve Days,” or “My Lady’s Lap Dog,” every Twelfth Day night. The party was usually a mixed gathering of juveniles and adults, mostly relatives, and before supper—that is, before eating mince pies and twelfth cake—this game and the cushion dance† were played, and the forfeits consequent upon them always cried‡. The company were all seated round the room. The leader of the game commenced by saying the first line. […] The lines for the “first day” of Christmas was said by each of the company in turn; then the first “day” was repeated, with the addition of the “second” by the leader, and then this was said all round the circle in turn. This was continued until the lines for the “twelve days” were said by every player. For every mistake a forfeit—a small article belonging to the person—had to be given up. These forfeits were afterwards “cried” in the usual way, and were not returned to the owner until they had been redeemed by the penalty inflicted being performed.
Alice Gomme (1898). The Traditional Games of England, Scotland, and Ireland, volume 2, p. 319. London: David Nutt.
† For “cushion dance” see volume 1, pp. 87–94. ‡ For “crying forfeits”, see volume 1, pp. 137–138.
Readers ought to recognise this game as resembling other cumulative memory games, such as the 20th century “I went to the zoo and I saw …” This origin adequately explains why the presents are repeated on each day of Christmas, without needing to go into the question of whether twelve partridges were sent, or only one.
“The Twelve Days of Christmas” is just one of a group of similar cumulative recitations:
In Scotland, early in the nineteenth century, the recitation began:
The king sent his lady on the first Yule day,
A popingo-aye [parrot];
Wha learns my carol and carries it away?
The succeeding gifts were three partridges, three plovers, a goose that was grey, three starlings, three goldspinks, a bull that was brown, three ducks a-merry laying, three swans a-merry swimming, an Arabian baboon, three hinds a-merry hunting, three maids a-merry dancing, three stalks o’ merry corn. In the Cambrésis, in the north of France, the game is called ‘Les dons de l’an’, and the sequence is one partridge, two turtle doves, three wood-pigeons, four ducks flying, five rabbits trotting, six hares a-field, seven hounds running, eight shorn sheep, nine horned oxen, ten good turkeys, eleven good hams, twelve small cheeses. In the west of France the piece is known as a song, ‘La foi de la loi’, and is sung ‘avec solemnité’, the sequence being: a good stuffing without bones, two breasts of veal, three joints of beef, four pigs’ trotters, five legs of mutton, six partridges with cabbage, seven spitted rabbits, eight plates of salad, nine dishes for a chapter of canons, ten full casks, eleven beautiful full-breasted maidens, and twelve musketeers with their swords. A Langeudoc chant is similar, but the gifts are made on the first fifteen days of May.
Iona Opie and Peter Opie, eds. (1951). The Oxford Dictionary of Nursery Rhymes, pp. 122–123. Oxford: Clarendon Press.