That wouldn't be a bad idea if it could be done -- I'm not sure it can because I don't think the jetbridge can reach the rear door. Some LCCs do use both doors to board narrow bodies (using stairs rather than jetbridges). It greatly accelerates boarding and shortens the turnaround time.

You might think it's silly to do this on a narrowbody because it's a small plane but in fact there are two reasons why it's actually more important on narrowbodies rather than wide bodies.

a) Narrowbodies have a single aisle. So there are actually more seats boarding per row length of aisle on a 320 (6) than there are on a 777 (4.5). To make matters worse people sometimes cross over or pass their bag across on the widebody and that helps alleviate bottlenecks.

b) Narrowbodies are used on shorter flights so they spend more of their time proportionally on the ground. Turnaround times are absolutely critical for LCCs where they're often on the order of 20-40minutes. BA on the other hand typically has nearly 2 hours scheduled between flights.

I suspect you're actually both wrong about the "A" and "B" markings. I suspect it's so that they can park two small planes at a single gate. American airports often have two flights boarding from a single gate. It's a terrible idea that often leads to passengers on the wrong flights (including more than once unaccompanied minors) but I assume it saves significant costs somehow.