How will installing Mavericks affect Parallels Desktop?

You will need to update Parallels to a newer version. I suspect you might have other software that needs updating as well. See this site to check for compatibility: http://roaringapps.com/

Parallels Desktop for Mac version 7 and earlier will not support OS X Mavericks (10.9). You will not be able to launch your Windows Virtual Machine or directly use your files through Parallels Desktop for Mac version 7 and earlier.

To continue using Parallels Desktop for Mac in OS X Mavericks you should upgrade to Parallels Desktop 9 for Mac: http://www.parallels.com/store/

You do not have to update from P8 it will work fine (I presently use it) with Mavericks. You can buy P9 but no required.