The Cold War was an unarmed conflict between the United States of America and the USSR.
The reason why it led to the growth of white-collar jobs after World War II is that the Cold War was more of a technological and ideological conflict.
One country trying to be more advanced and have better things to offer than the other. This means more investment on education, technology, etc.
That led to the growth of white-collar, intellectual jobs, rather than odd jobs, or manual labor.