On the CallHome dataset humans confuse words 4.1% of the time, but delete 6.5% of words, most commonly deleting the word "I".
Their ASR system confuses 6.5% of words on this dataset, but only deletes 3.3% of words, so depending on how you view this their claim about being better than humans isn't definitely true, if you consider the task to be speech recognition, rather than transcription.
Also, while the overall "word error rate" is lower than humans it's not clear if this is because the transcription service they used is not seeking perfect output, but rather good enough output and the errors the transcription service makes may not be as bad as the errors the ASR system makes in terms of how well you can recover the original meaning from the transcription.
It's clearly great work, but reaching human parity is marketing fluff.