The Yang-Mills (YM) equation in three spacetime dimensions (3D) can be modified to include a novel parity-preserving interaction term, with an inverse mass parameter, in addition to a possible topological mass term. The novelty is that the modified YM equation is not the Euler-Lagrange equation of any gauge-invariant local action for the YM gauge potential alone. Instead, consistency is achieved in the "third way" exploited by 3D minimal massive gravity. We relate our results to the "novel Higgs mechanism" for Chern-Simons gauge theories.