The memory does not necessarily work fine on the Mac, in fact it doesn't at all, Parallels is a OSX application, what happens is that Parallels uses quite a big chunk of memory and as such it is likely encountering a defect present in the new RAM module, unfortunately this may indeed happen on a newly purchased module. This will also likely happen if you fill up all the RAM on the Mac (dozens of browser tabs and windows, Photoshop, etc, etc).
What you need to do is test the memory via hardware diagnosis, like memtest or Apple own hardware diagnosis tools: http://support.apple.com/kb/HT1883
This is also what Apple will recommend you to do and what any technician should recommend. In fact this should be the first thing anyone should do when they upgrade their RAM and any technician worth they salt should know that (this is a direct reference to Parallels Support). The importance of this is to catch defects early while the RAM warranty is still valid and to be sure any future problem is not related to this.
Last edited: Jan 31, 2013