The AMMCLib User's Guide describes how the continuous-time PI controller can be discretized to obtain the difference equation suitable for implementation in the digital microcontroller.
The equations GFLIB_ControllerPIp_Eq3 and GFLIB_ControllerPIp_Eq4 describe the calculation of the discretized proportional and integral part independently in parallel, thus the name 'parallel form'.
The equation GFLIB_ControllerPIr_Eq3 describes the calculation of the discretized proportional and integral parts together in one common recurrence equation, thus the name 'recurrent form' (also sometimes referred to as 'serial form').
Both forms are functionally equivalent. The parallel form is perhaps more convenient for manual tweaking of the P and I gain, since it allows setting them independently. The recurrent form is often used in industrial controllers and can be tuned with existing tuning tools which expect this form.