I agree, belt tensioner, or probably crap aftermarket alternator. If it was parasitic drain it would be 14 volts or more when running. I think everyone here has a story about non OEM alternators going prematurely bad or sometimes bad right out of box.
Our 2008 went through 2 batteries in 3 months. Turned out to be the battery cables were corroded not at the battery connections but further down. Worth a shot if you haven't changed those yet.
This part number started with 2006 E from what I can find. Does anyone know about 03-05 E's?Have you heard of the Honda Electronic Load Detector Unit (38255-S5A-003) ? Its in the under-hood fuse box and when my '09 voltage was fluctuating while driving from my dash mtd voltage gauge, Honda recommended I try that - FREE advise! My 6 yr old Honda battery was due, so i replaced it, and the NEGITIVE cable along with an Optima Yellow top. Since last fall its been perfect, so it was either the cable or the ELD. The ELD is in the under-hood fuse box, and it sounded harder than it was. Took about 30 min since mine was crazy, the underside hidden plug twisted-up, which took time to correct.
The ELD is the BROWN thing the Blue sharpie is touching and on the NEW negative battery cable I coated the bare wire section with liquid electrical tape so it won't corded like the OEM one shown - good enough for NASA, good enough for me.