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In this paper we obtain the following weighted $L^p$-type regularity estimates $$ B\left(\left|\mathbf{f}\right|\right)\in L^{q}\left(\nu, \nu+T; L^q_w(\Omega) \right) \ \mbox{locally} \Rightarrow B\left(\left|\nabla u\right|\right)\in L^{q}\left(\nu, \nu+T; L^q_w(\Omega) \right)\ \mbox{locally} $$ for any $q>1$ of weak solutions for non-homogeneous nonlinear parabolic equations with Orlicz growth \begin{equation*} u_t-\operatorname{div} \left(a\left(\left( A \nabla u \cdot \nabla u\right)^{\frac{1}{2}} \right) A \nabla u \right) = \operatorname{div} \left(a\left(\left|\mathbf{f}\right| \right) \mathbf{f}\right) \end{equation*} under some proper assumptions on the functions $a, w, A$ and $\mathbf{f}$, where $B(t)=\int_0^t \tau a(\tau)\,d\tau$ for $t\geq 0$. Moreover, we remark that two natural examples of functions $a(t)$ are $$ a(t)=t^{p-2} \quad\mbox{($p$-Laplace equation)}\quad \mbox{and}\quad a(t)=t^{p-2 }\log^\alpha \big( 1+t\big) \quad \mbox{for}~\alpha> 0. $$ Moreover, our results improve the known results for such equations.