Uniformly Most Powerful (UMP) centralized detection for the composite binary hypothesis problem has been well researched. This paper extends the UMP methodology to the parallel distributed detection problem, when the local observations are independently distributed. A collection of general theorems and corollaries define sufficient conditions for the existence of a UMP parallel Distributed Detection (UMP-DD) under one set of fusion rules. These same conditions under another set of general fusion rules result in at least a Locally Most Powerful Distributed Detection (LMP-DD) rule. The subtleties of these conclusions are explored using informative examples that highlight the strengths of this approach and introduce new groups of UMP-DD tests.