Mazda 3 2004: Parking lights turns off when Headlights are o

I have a weird issue with my 2004 Mazda 3. If I turn on the headlights on the combination switch, everything works normally for a few minutes. That is, the headlights turn on, the parking/tail/license plate lights as well, and the dashboard lights dim.

However, after a few minutes of driving, the parking, tail lights and license plate lights all go off, leaving on the headlights on. Playing with the combination switch back and forth from on to off does not solve the problem. The only way to get the parking lights to light up again is to leave all lights in the off state for a few minutes, and then try turning them on again. But the same problem will arise again after a few minutes of leaving them on.

Note that if I leave only the parking lights on (without the headlights), I never have any issue. It only occurs when the headlights are on as well.

I have tried the following in order to find the source of the problem but with no luck:

1. I have replaced the relay for the parking lights within the internal junction box. This did not solve the problem. In fact, when the problem arises, the relay stops clicking when turning the parking lights on, which could indicate that power is not even being sent to the relay.

2. I have inspected the fuses and even swapped them with similar fuses within the junction box. This did not change anything. I have also verified that the fuses are of the correct specification.

3. I have checked the wattage of all the bulbs (headlights, parking lights, tail lights, license plate lights) and they all seem to be according the specification with the exception of the front parking lights which are 5W bulbs instead of 8W bulbs. I have tried running the car with them disconnected, but the problem still arises.

The problem could be with the combination switch. But since the dashboard lights dim normally even when the parking lights are not working, I believe that the switch is working properly. But I could be wrong since I do not know exactly how the switch works.

How do I diagnose this problem? I have no idea what to check next.