Monotone conservative difference schemes of the second order of local approximation in space for a differential problem with boundary conditions of the third kind on arbitrary nonuniform grids in both time and space are constructed. For the multidimensional parabolic equation, monotone local one-dimensional difference schemes of the second order of local approximation in space on the minimal stencil are proposed. A priori estimates in the norm \(C\) are obtained on nonuniform grids by means of the grid maximum principle. Two numerical experiments are presented.