Because his experience in the naval medical corps had been rewarding, Bob applied to medical school after he (was discharged) from the navy.

The answer is No Error, but shouldn’t the parenthetical phrase be changed to “HAD BEEN discharged” since it is with “after,” indicating a past perfect tense?

Nope. The “after” does all the work that a past perfect tense might otherwise do. Because you have the “after,” you already know all you need to know about the order of events, obviating the need for past perfect tense.