In this paper, we investigate continuity properties of the efficient solution map of a parametric nonlinear multiobjective optimal control problem. First, by using the equimeasurability condition of the admissible control set, we obtain the compactness and arcwise connectedness of the feasible solution set. Next, we suggest new concepts of the quasi-arcwise connected integrand and employ them to study the semicontinuity of the efficient solution map of this problem. When the multiobjective function does not satisfy these conditions, we propose an estimation hypothesis for approximate efficient solutions to address lower semicontinuity conditions of the efficient solution map of the reference problem. To illustrate the applicability, we apply the obtained results to two practical models, including Glucose model and Epidemic model.