After installing aftermarket tail lights on my 94 Honda Prelude Si, the broken lamp light and hand brake (!) Dash lights stay on with the ignition switiched on. All of the tail lights work and the dash lights were fine before the swap.
Did you install LED lighting instead of incandescent lighting? LEDs don't draw as much current as incandescent, so your Honda may think (because of the low current draw) that you have a failed taillight bulb. You may have to intentionally add extra current load at the taillights to make that dash light go away.
The hand brake light, though, would be a totally different issue. THAT should be controlled only by a switch on the hand brake.