You've connected the power to your neon lights through the dimmer instead of the parking light wire. When you adjust the interior light the voltage in the dimmer either goes above 12v or below 12v depending. When you turn your interior lights down, probably at least half the dimmer voltage is probably around 10.5v which causes the gauges to flash. Instead just hook your gauges to your parking light wire and you should be fun. What year is your car? Almost all the honda's just go to the fuse panel and with a voltimeter test the larger gauge wires coming from it to determine the parking light wire and tap into it. Don't just jam a wire into the fuse panel. That is dangerous, it can damage the fuse panel or cause a fire over time.
-Mike