Let $$(M^{n},g,e^{-f}dv)$$ be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation $$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$where $$(x,t)\in M\times (-\infty ,\infty )$$ and a, $$\alpha $$ are arbitrary constants. Under the assumption that the $$\infty $$-Bakry-Emery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation.