  A linearized numerical scheme is proposed to solve the nonlinear time
fractional parabolic problems with time delay. The scheme is based on the
standard Galerkin finite element method in the spatial direction, the
fractional Crank-Nicolson method and extrapolation methods in the temporal
direction. A novel discrete fractional Gr\"{o}nwall inequality is established.
Thanks to the inequality, the error estimate of fully discrete scheme is
obtained. Several numerical examples are provided to verify the effectiveness
of the fully discrete numerical method.
