Be sure that the dash light brightness control switch is not turned off.
Yes, your tail lights will go out when your dash board lights go out.
the dash and radio lights are supposed to dim when the headlights come on. Make sure you have the brightness of the dash lights turned up as much as possible.
turn the lights on and make sure the dimmer is not turned all the way down
Check your fuses
It would go to the dimmer that controls the brightness of the dash lights.
You could have a blown fuse, these lights may also have a knob to dim them make sure this is on.
Just guessing without having it in front of me but, I would suspect the roller dimmer switch on the dash. I know that if it is unplugged, all dash lights go off. So check the plug-in for a short.
your fuse is blown. car manufacturerers make the dash lights go off when the tail light fuse blows so you know it is blown.
Short
Check for a loose ground wire - sounds like lights are backfeeding searching for ground
might be a fuse
check the fuses