How can I use one of the documents preview icon as the parent folder's icon in OS X?
As far as I know, it's not possible to do so dynamically. 

What you can do is change the icon manually (but then you have to do it for each folder of course...).

Open your icon with Preview. Then copy the image by using the build in copy/paste functionality. Select the whole image with cmd+ A and copy it with cmd+ C.

Then open the 'Info' window (right-click and Get Info) of the folder and select the icon on the upper left corner. If you paste cmd+V the image, you'll see the folder icon got replaced by the image. 

And of course, there is an app for that.