The U.S. did not have parity with the USSR after WWII, but absolute superiority. By the times of the first Soviet nuclear test in 1949, the U.S. had a stockpile of about 200 bombs. During the first four years after the war, the U.S. could have, if it had wanted, dictated terms to any nation in the world--including the USSR. America could have forbidden any other nation to have the bomb, and enforced the ban with the threat of nuclear attack. The fact it had already shown its willingness to use atom bombs would have made that threat pretty credible.