We establish Strichartz estimates for a generalized Hermite--Schrödinger equation associated to a family of differential-difference operators involving the Dunkl Laplacian and unbounded potentials. This family includes the Hermite and Laguerre differential operators in particular. The study relies on the analysis of the so-called (k,a)-generalized semigroup studied by Ben Said-Kobayashi and Orsted. Moreover, we prove that homogeneous Strichartz estimates for the Schrödinger equation associated to the Dunkl Laplacian can be obtained from those for the generalized Hermite--Schrödinger equation.