I have a 2005 Corolla. The green daylight indicator only comes on when I turn the regular low beams on. It does not appear when only the daylights are on. Sometimes, I don't even see a difference when I switch from my daylights to my low beams. Also, in order for my parking lights to work, the car engine has to be completely off, with only the accessory key on. Is this normal?

Yep! Normal. I have a 2005 Corolla and it works the same way you described.

Hi
That's absolutelynot normal. I suggest you get someone with auto-electric knowow to check yourignition and combinaton switches.

I would take a look at the switch and check for abnormal rating. get a wiring layout.