I think you could argue that there is a lot of a propaganda conflict in the Cold War. The United States and the Soviet Union were trying to out-psych each other. I heard recently that the "We will bury you" comment was actually mistranslated.
In the cold war, the US and Soviets were constantly threatening one another. Soviet premier Nikita Khrushchev’s famous line was actually mistranslated.
Khrushchev gave a speech in which he uttered a phrase that interpreted from Russian as "we will bury you." ...However, the translation was a bit too literal. The sense of the Russian phrase was more that "we will live to see you buried" or "we will outlast you." (theweek)
Nonetheless, there was a lot of propaganda on each side. Perhaps the Soviets appreciated seeming to be tougher than they meant to be. Either way, it was one of those things that escalated and could not be taken back, because it led to an even bigger arms race.
The other day, you asked if the Cold War was a military conflict. The answer to this question is pretty much the same as for that question. The Cold War was partly, but not solely, a propaganda conflict.
We can say that the Cold War was partly a propaganda conflict because both sides wanted to try to convince the rest of the world that their way was the best. The West and the Communist Bloc were trying to convince the rest of the world (as well as, to the extent possible, the citizens of the other side) that their way was the best. Therefore, we had propaganda like the Voice of America broadcasts that were beamed into the Soviet bloc. The Soviets also put out propaganda, such as their attempts to convince Third World people that the US was fundamentally a racist society. The “Kitchen Debate” between then-vice president Nixon and the Soviet leader Khrushchev was another example of propaganda where each side was trying to portray its way as best.
Thus, propaganda was certainly one aspect of the Cold War as both sides tried to win the approval of outsiders.