  Let $(W,S)$ be a Coxeter system with $I\subseteq S$ such that the parabolic
subgroup $W_I$ is finite. Associated to this data there is a \textit{Hecke
algebra} $\scH$ and a \textit{parabolic Hecke algebra}
$\scH^I=\mathbf{1}_I\scH\mathbf{1}_I$ (over a ring $\ZZ[q_s]_{s\in S}$). We
give a complete classification of the commutative parabolic Hecke algebras
across all Coxeter types.
