  We study the Li--Yau inequality for the heat equation corresponding to the
Dunkl harmonic oscillator, which is a non-local Schr\"{o}dinger operator
parameterized by reflections and multiplicity functions. In the particular case
when the reflection group is isomorphic to $\mathbb{Z}_2^d$, the result is
sharp in the sense that equality is achieved by the heat kernel of the classic
harmonic oscillator. We also provide the application on parabolic Harnack
inequalities.
