Can post-doc inquiries be parallel?
You should apply all at once. Everyone else will be, and professors and labs know that you will apply to multiple jobs at the same time. Once you accept a postdoc somewhere, you should immediately withdraw your other applications that are still pending. You may find yourself needing to get an extension of time from one organization that has made you an offer while you wait on your pending offers. Hopefully, you will be able have several offers to consider at once in order to make the best choice, but you may have to decline an offer if better ones are probably coming down the line and an extension of time isn't available.