Hello Jaloring,

A shorted alternator diode could slowly drain the battery away (by not able to properly charge the battery during each use).

See:
http://www.aa1car.com/library/battery_runs_down.htm

You'd need a voltmeter to test for this. Or you could bring the car to a alternator repair shop to diagnose. If you choose to do the first, set the voltmeter to AC (alternating current reading), engine running, one probe to the B+ terminal on alternator and another probe to ground (alternator body). Probe polarity doesn't matter as AC goes both ways.

The reading should be less than 0.5volts. If more than that one or more of your diode(s) may have shorted.
Yes the car could still run with parasitic AC in the system but the delivered voltage going to be less than 14volts leading to undercharged battery (plus low reading under load at the in-dash voltmeter).

Hope this helps,
Amarin.