We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half‐plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The paper is a further development of earlier work by the authors, where we treated the homogeneous equation in a Hilbert space framework. Special attention is given here to the treatment of the forcing term. The method is combined with finite‐element discretization in spatial variables.