The phrase 'a partridge in a pear-tree' is a corruption of the Latin words 'apertuit in aperto', which mean 'she gave birth in the open'...ie in an exposed place, the stable. When medi�val monks chanted these words in church, the native peasantry in the congregation misheard them as the phrase we are now familiar with. There is, therefore, no real 'symbolism' involved at all.
The original lyrics involved things of much more religious significance. One was the baby Jesus, two was Mary and Joseph, but I'm not sure about the rest.
I wouldn't be too sure of the Latin QM ! It could be either 'parturit in aperto' or 'parturit aperte'. The first in classical Latin means in the wide open spaces, the wide outdoors. The second means 'gave birth without secrecy, openly, without seeking to hide, unashamedly' My vote is for the second! Mary was not in an open field but she certainly did not seek to dissemble or hide her mysterious conception and the birth It would read better if the word order was reversed 'aperte parturit' but as none of the rest of the lines are anything like Latin it may be a fanciful schoolboy's or non-classicist's supposed 'explanation', not the church's words.