This one's a doozy. Don't say we didn't warn you.
Let's just set it all out right at the beginning: The Butter Battle Book is a pretty clear allegory for the Cold War arms race. But what's great about allegories is that they're flexible and fuzzy. That means that while the Good Doctor may have intended for his book to allegorize the Cold War, we can think of it in terms of current arms races, the supercomputer race, or any other adversarial relationship between two groups based on technological advancement.
But since we're in Seussland, let's think about what Dr. S might have said. We'll break it down together, piece by piece.
If we cut to the chase, Grandpa is probably meant to represent the United States Military during the Cold War. He's a general—remember the fancy hat—and because this is a kids' book, it's best to use him to stands in for the entire army, instead of complicating things with too many characters.
Grandpa's strategy? Containment, it seems. After all, he never does more than threaten to use his weapons:
In those days, of course,
the Wall wasn't so high
and I could look any Zook
square in the eye.
If he dared to come close
I could give him a twitch. (30-35)
This guy wasn't crossing into Zook territory. No, he was just putting up a tough front to keep the Zooks "contained." Grandpa's strategy matches the United States', which, in the Cold War, intended to hold the Soviet Union to a restricted geographical region.
VanItch, on the other hand, is Grandpa's counterpart: he represents the Soviet Union's military and command. And yes, he gets a fancy hat, too.
So we've got the military all sorted out, but who's in charge? Well, we'd argue that in the case of the Yooks, the Chief Yookeroo represents the President of the United States (Roosevelt, Truman, Eisenhower, Kennedy, and especially Reagan).
This particular commander-in-chief gets to watch the race from afar. While we're pretty sure no American president has ever treated such weighty affairs as a game, Dr. Seuss does point to a potential danger when the leader doesn't have to be in the trenches. The Chief also seems to benefit from the "international" tensions at hand and pushes the nation into an adversarial relationship. We'll let you decide what Seuss was trying to say about his current president.
As goofy as these Seussy weapons they might seem, they have their roots in real-life Cold War weapons.
The Wall starts off a wee thing that Grandpa Yook can see right over. But by the end of the story, it towers above the young Yook and his gramps. Sure, the wall might be metaphorical, separating these two like groups—Yooks and Zooks—from a friendly relationship, but it almost certainly reads as a symbol for the Berlin Wall.
Quick and dirty rundown of the Berlin Wall: The East Germans, allied with the Soviet Union, built the Wall during the Cold War. The intention, they said, was to keep out the fascists they believed still ran rampant in West Germany. The Wall was finally torn down in 1990, marking the symbolic end of the war.
So is Seuss suggesting that the Yook-Zook wall should be torn down? Is that all it will take to stop the arms race? Or is the Wall just an effect, and the arms race needs to end before it can be taken down?