  We investigate Laplace type and Laplace-Stieltjes type multipliers in the
$d$-dimensional setting of the Dunkl harmonic oscillator with the associated
group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context
of Laguerre function expansions of convolution type. We use Calder\'on-Zygmund
theory to prove that these multiplier operators are bounded on weighted $L^p$,
$1<p<\infty$, and from $L^1$ to weak $L^1$.
