A two-grid algorithm is presented and discussed for a finite volume element method to a nonlinear parabolic equation in a convex polygonal domain. The two-grid algorithm consists of solving a small nonlinear system on a coarse-grid space with grid size $H$ and then solving a resulting linear system on a fine-grid space with grid size $h$. Error estimates are derived with the $H^1$-norm $O(h+H^2)$ which shows that the two-grid algorithm achieves asymptotically optimal approximation as long as the mesh sizes satisfy $h=O(H^2)$. Numerical examples are presented to validate the usefulness and efficiency of the method.