It is important to understand the variations on the PID algorithm when tuning and when choosing a version that is consistent within your use context.  Unfortunately, there are many names for the several key versions of the PID algorithm, and names such as series or parallel are each applied to two distinctly different PID formulations.  

To help unify the nomenclature, ISA has a Standards Committee 5.9 that is suggesting naming conventions for the PID algorithms.  This article follows what is currently on the agenda to be proposed, but also reveals some of the commonly applied alternate names that you may be using.