<p>In this paper, we investigate the numerical solution of linear parabolic equations using continuous Galerkin methods in both space and time. We analyze the numerical errors of the space-time Galerkin method, identifying the leading and higher-order terms that influence the error profile, which helps us locate space-time superconvergence points. Based on this error analysis, we propose a postprocessing technique to enhance the accuracy of the Galerkin solution and construct a corresponding a posteriori error estimator. Numerical results demonstrate that the postprocessing improves the approximation accuracy of the Galerkin solution by one order in both time and space, and the technique is also effective for nonlinear parabolic equations. This study offers a comprehensive framework for understanding and improving error behavior in space-time continuous Galerkin discretizations of parabolic equations.</p>