Yes. You can upgrade at any time, and just pay the difference in list price between the two Editions.
Essentially, a "part" is one line in the parts list, regardless of how many copies you need of that item. Here is an example to illustrate what the terms mean. Consider a (very) simple table with a top and four legs. When entering this into CutList Plus, there would be just two parts, called "top" and "leg". You would have one copy of the top part, and four copies of the leg part.
If you aren't sure which to get, we recommend you start with the Gold Edition, which is the most popular. If you find that you need the features in the Platinum Edition, you can always upgrade later for just the difference in cost.
CutList Plus runs on Windows XP, Vista and Windows 7, 8 and 10. Here at Bridgewood Design, we run CutList Plus on our Macs using Parallels with Windows 8, but you can also use Fusion or Boot Camp. There is no Mac OS version of CutList Plus.