This paper studies the prescribed-time output-constrained tracking control problems for Mars entry vehicle systems. By using the properties of the time-varying high-gain function and output-constrained conditions, time-varying barrier Lyapunov functions are constructed. Based on such Lyapunov functions, time-varying adaptive controllers are designed to solve the considered problem. It is proved that the proposed controllers can drive the errors converge to zero in a prescribed time under unknown disturbances, and the control signals are bounded. Both the symmetric and asymmetric cases are considered. Finally, the effectiveness of the proposed methods is illustrated by several numerical simulations.