Alarm system problem with a 2005 Pilot

[Kero.Of.Spirit]

Hello!
I need some help with a problem my parents are having with their used 2005 honda pilot. They bought it from a private owner last week, and so they only have one remote control key (with the lock and unlock buttons). my mother has an old fashioned key with no controls on it, just the key itself with the honda logo on top. Whenever she uses this key to open the driver side door, the alarm goes off as soon as she opens it. It does not do this when she opens any of the other doors with her key, just when she [U]unlocks[U] the drivers door with it. can anyone tell me how to fix this problem? its becoming increasingly frustrating for my parents.

Also, no matter what we do, when we turn they car off one or the other of the lights in the backseat will stay on for several hours, and only with alot of unlocking and re-locking the doors can we make them both turn off. does anyone know what a possible reason for this may be?