The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the nth power of the iteration matrix derived via a fourth-order compact $\theta$-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.