This depends on the specifics of the code and the errors being thrown.
With the first way, say processed_data = big_process(parsed_data) fails. output = postprocess_result(processed_data) will still run since they're separate checks. Also, all three failures have the potential to be handled in three different ways.
With the second way, say the first line fails. The third line will not run since execution is transferred to the except block. Also, all three failures will be handled the same way (unless you manually use flags or some other mechanism to differentiate).
So the answer is, neither is really more Pythonic. The pieces of code are not equivalent, so you should use whichever is more appropriate for your particular case.