I describe an adaptive h-refinement method for solving systems of parabolic differential equations in three space dimensions on hexahedral grids.  These grids typically have irregular (hanging) nodes.  Solutions are calculated using Galerkin's method with a piecewise trilinear basis in space and a BDF code in time.  New a posteriori error indicators based on interpolation error estimates for irregular grids are used to control refinement and coarsening.   A more efficient algorithm for assembling banded portions of the Jacobian is introduced.  A simple strategy for dealing with storage limitations by limiting the level of refinement is developed.  Computational results demonstrate the effectiveness of the adaptive method on linear and nonlinear problems.