In this work we present an efficient \emph{a posteriori} output error bound for model order reduction of parametrized evolution equations. With the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system, the output error bound can be estimated sharply. Such an error bound successfully avoids the accumulation of the residual over time, is a common drawback in the existing error estimation for time-stepping schemes. The proposed error bound is applied to three kinds of problems. The first one is the unsteady viscous Burgers\' equation, an academic benchmark of nonlinear evolution equations in fluid dynamics often used as first test case to validate nonlinear model order reduction methods. The other two problems arise from chromatographic separation processes. They are batch chromatography with (nonlinear) bi-Langmuir isotherm equations, and continuous simulated moving bed chromatography with linear isotherm equations, where periodic switching is involved. Numerical experiments demonstrate the performance and efficiency of the proposed error bound. Optimization based on the resulting reduced-order models is successful in term of accuracy and the runtime for getting the optimal solution.