Here is some real world advice:. Do NOT listen to US government websites such as the one posted above. Most of the time (not always) stuff lie that is based upon perceptions and average anomolies that happen given the right circumstances in ANY country. That includes the US of A. Now, traveling with $10k USD cash is risky in the USA and everywhere.... including a US casino. Is it in any different in PY? Hell no! It's the same exact risk. Paraguay is safe for the most part and for the most part, you will be just fine. Don't listen to some know it all's who claim otherwise. You can walk through downtown Asuncion at 2am with $10k in your pocket and be just as safe as walking through downtown Jacksonville Florida at 2am with $10k in your pocket. And probably a little safer cause ASU is a ghost town at 2am on a weeknight. No one under normal circumstances will Rob you or approach you or even talk to you. So, that's it in a nutshell.
Don't carry $10k on you if you don't have to, but certainly don't be "scared" any more than you would be if you were in the USA or any other major country. Don't read propoganda websites which proclaim absolute knowledge of a place either.