How does parallel mode differ from serial mode in the Moog One's filter section?

Each synth layer has two filters and the individual oscillators can be assigned to either or both of them. If both filters are activated, the ROUTE button selects whether they are routed in parallel or serial mode. 

PAR routing means that the two filters operate separately, side by side. In this mode the MIX knob acts as a simple mixer with the State Variable Filter on one side and the Ladder Filter on the other. 

SER routing means that the output of the SVF feeds into the input of the Ladder Filter, forming a continuous “filter chain.” In this mode, the MIX knob will only let you hear the SVF filter when it is rotated fully counter-clockwise. As you rotate it to the right, you will hear more of the Ladder filter blended in, but because the Ladder follows the SVF in the signal path, you will still be able to hear the effect of the SVF even when it is fully clockwise. 

Furthermore, the SVF itself is actually two filters set at a fixed distance from each other. And by pressing the MORE window in the filter section you can adjust whether those two filters run in parallel or series routing.

NOTE: Although sharing similar attributes, this parameter has a different function than the SER / PAR ROUTE button on the Front Panel. That button specifies whether the SVF and Ladder Filter are running in Series or Parallel. The SVF SERIES/PARALLEL selection only affects the elements inside the SVF module