Risk II on a modern day Macintosh
To the OP:
What version of OS X were you using to play Risk II in?
10.?
and on what hardware?
and with what version of Risk II?
I have done a bit of research on your behalf.
It seems Risk II was released in 2000, which should mean that it was written for Mac OS 9 or less. I cannot seem to find a version greater than 1.0.x (although it is possible that they wrote an OS X version and kept it within the 1.0.x numbering framework, which is a strange decision).
That would make it subject to being run in the Classic Environment in OS X, which only ran through Tiger, 10.4. The Classic Environment was eliminated in Leopard 10.5.
However here is a post suggesting he ran it in Leopard (10.5) with some graphics/colors problems:
https://forums.macrumors.com/threads/634625/
And here is a post suggesting these problems continued into Snow Leopard:
https://forums.macrumors.com/threads/812707/
So if it runs in Leopard and Snow Leopard, there must be an OS X version of Risk II (since the Classic Environment was eliminated in Leopard).
Here is a screenshot of Risk II performing well in Snow Leopard:
[see first attachment]
But here is another screenshot of the similar color problems running in Snow Leopard on different hardware:
[see second attachment]
And similarly, here is a screenshot of similar problems with Snow Leopard in Parallels 7 in Lion:
[see third attachment]
So if we can figure out why it sometimes works in Snow Leopard, then maybe we can get it to work with Snow Leopard installed in Parallels 7 in Lion...