Yes the only place where 12 volts and the tail light system are close together is in the umbilical cord. First open the male plug, if it is a standard 7 pin plug, and check for connection between the 12 volt
line and the right parking light wire. Looking at the back of the plug while it is your hand the 12 volt
connection will be at 11 o clock and the right tail light will be at 9 o clock.
If this is not the problem there may be a short in the umbilical cord itself, not uncommon with Older rigs. My cord had a nick in the outer jacket and water got in and opened the brake line connection through corrosion. Replace the cord.