Was the Southern Hemisphere cooler during the Younger Dryas?
According to the paper The Younger Dryas Climate Event (Carlson, 2013), the Younger Dryas was a period of warming in the extratropical Southern Hemisphere, and quite a significant one as well, as, from the article:


  SST records show a warming
  of 0.3–1.9 C from the southeast Atlantic to New Zealand


This is supported by


  Speleothem and pollen records from New Zealand and
  pollen records from South America confirm that the Younger
  Dryas was generally a period of warming


Citing glacial retreat measured in places such as New Zealand, the article Glacier retreat in New Zealand during the Younger Dryas stadial (Kaplan et al. 2010) hypothesised the mechanism to be due to


  extensive winter sea ice and curtailed meridional ocean overturning in the North Atlantic led to a strong interhemispheric thermal gradient8 during late-glacial times, in turn leading to increased upwelling and $\mathrm{CO}_2$ release from the Southern Ocean, thereby triggering Southern Hemisphere warming during the northern Younger Dryas.


Research on the deposits observed in the Australian tropics of the age in the article Late Pleistocene record of cyclic eolian activity from tropical Australia suggesting the Younger Dryas is not an unusual climatic event (De Deckker et al. 1991) revealed that the southern tropics experienced a drying phase, that fits well with a cyclic climatic phase.

In terms of the southern tropics, the article Tropical climates in the game of two hemispheres revealed by
abrupt climatic change (Ledru et al. 2002) suggest that there were significant variations, as recorded in pollen samples from the South American tropics, leading the authors to suggest that


  One consequence of the Younger Dryas changes would be the location of the Intratropical Convergence Zone in a southern position, so that even tropical regions would have been under Arctic influence.