Draftsight FAQ -
DWG and DXF files are versioned, which versions can DraftSight read and write?
Via the ODA libraries noted above as of October 2011, DraftSight can read .dwg and .dxf file versions as early as 2.5 and can save to any version from R12 to 2010.
Based on that I would say it is unlikely the 2013 version is supported. I have been meaning to install the latest Draftsight but probably won't for another few months. I can report back when I do. Also, if you actually go to the ODA website it looks like they have info available for the 2013 version, so if it is not yet supported in the latest version of Draftsight it could be eventually.
DraftSight allows users to access DWG/DXF files, regardless of which CAD software was originally used to create them, except AutoCAD 2013 and later... ...As stated on its home page, DraftSight uses Teigha Classic for .dwg files and Teigha for .dwg files specifications from the Open Design Alliance to interpret the dwg-file format.