The initial-boundary value problems for parabolic equations with nonlocal conditions have been widely applied in various fields. In this work, we firstly build an implicit Euler scheme for an initial-boundary value problem of one dimensional parabolic equations with an integral two-space-variables condition. Then we prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, the formulas used to approximate the solution derivatives with respect to time and spatial variables are presented, and it is proved for the first time that they have superconvergence. In the end, numerical experiments demonstrate the theoretical results.