Twelve Days of Christmas

The king sent his lady on the first Yule day,
A *papingo-aye;
Wha learns my carol and carries it away?

The king sent his lady on the second Yule day,
Three partridges, papingo-aye;
Wha learns my carol and carries it away?

etc.

Obviously a *papingo-aye (a peacock) doesn't work in the version now sung.
Nor does another version, My lady's lap dog.

From Alice B. Gomme, 1894-98, The traditional Games of England, Scotland, and Ireland

I think the 'pun' idea is incorrect, but perhaps others can find some weight to it. There doesn't seem to be any such indication in the UK rhymes.