In this work, we introduce the new notion of bi-parametric parabolic potentials in the framework of the Dunkl-Fourier transform which extends the classical bi-parametric potentials (for \(k=0\) ). Namely, we define the families of operators \(H^{k}_{\alpha ,\beta }:=\left( \dfrac{\partial }{\partial t}+(-\triangle _k)^{\frac{\beta }{2}}\right) ^{-\frac{\alpha }{\beta }}\,\,\,\,\hbox {and}\,\,\,\, \mathcal{H}^{k}_{\alpha ,\beta }:=\left( I+\dfrac{\partial }{\partial t}+(-\triangle _k)^{\frac{\beta }{2}}\right) ^{-\frac{\alpha }{\beta }}\,\,\,(\alpha ,\,\beta >0),\) where \(\triangle _k\) is the Laplace-Dunkl differential operator and I is the identity operator. Also, some properties of these parabolic potentials in the special weighted \(L^p_{k,0}(I\!\!R^d\times I\!\!R)\) -spaces are established.
