Don't forget: the nationalism in both the Yook-Zook rivalry and the Cold War was complimented by a desire for technological superiority. A dangerous cocktail, indeed.
Let's take a look at the Seuss version. What's happening here? Basically, each side is trying to one up the other in terms of modernization. When Chief Yookeroo has the Back Room Boys cook up a "newfangled kind of a gun" (78), he does so because the Zooks demonstrated superiority in the slingshot department. Yep, the need for a new gun is the direct result of VanItch's Jigger-Rock Snatchem.
Each time the Yooks or Zooks introduce a new technology, the other rushes to catch up and gain dominance. And it only stops when they reach "a gadget that's Newer than New" (165). How can you top that?
The Cold War arms race arrived at its climax at about the same position. Both the United States and the Soviet Union had nuclear weapons and in sufficient quantities that the results of a nuclear war seemed to be mutual annihilation. The climax? The Cuban Missile Crisis, when the United States and Soviet Union were poised like Grandpa and VanItch in the final lines of the book. At that moment, people became like the young Yook asking, "Be Careful! Oh, gee! / Who's going to drop it? / Will you...? Or will he...?" (200-202). To see how that turned out, check out Shmoop's own take on The Cuban Missile Crisis. Although, since we're all still here, you can probably guess.