We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $p$-Laplacian type with $p>\frac{2n}{n+2}$ in weighted Lebesgue spaces $L_w^q$. We introduce a new condition on the weight $w$ which depends on the intrinsic geometry concerned with the parabolic $p$-Laplace problems. Our condition is weaker than the one in ^[13], where similar estimates were obtained. In particular, in the case $p=2$, it is the same as the condition of the usual parabolic $A_q$ weight.