You can continue to run SharePoint Workspace 2010 with an installation of Office 2013. In fact, the easiest update path for installing Office 2013 is simply to leave Office 2010 in place. You can keep both Office versions on your system and run them side by side.
However, some people prefer to remove older versions of software from their computers, if they think they won’t use them anymore, or perhaps to free up space. If this is what you want to do, then uninstall Office 2010 (except for SharePoint Workspace 2010) before you install Office 2013.
For details on removing Office or Office components from your system, see Install Office 2010.
What if I’ve already installed Office 2013 and now I want to remove Office 2010?
If you’ve already installed Office 2013, you can still remove Office 2010 from your computer. When you’re done removing Office 2010, you must repair your Office 2013 installation. For details on repairing your Office installation, see Repair or remove Office.