[BLE 'BURBAN]

What is the year and model of your vehicle? You also failed to mention if you are using standard bulbs or if you are utilizing any L.E.D.'s

So lets theorize (I never ASSuME anything) that you are using standard bulbs and not L.E.D.'s

The side marker lights are not grounded in the usual fashion.They have in fact 2 power leads going to them. The brown lead is your marker/parking light circuit and the Blue wire runs from your front turn signals to the side marker/parking light. When you turn on your turn signals the side marker/parking light flashes alternating with the front turn signal light in a wig-wag style.
If your Right side indicator light is lit up on the dash and you have no right side turn signal, then you have either a bad bulb in the front or rear of the vehicle (Most common problem) or a bad ground on that side.
I could also be a corroded socket or broken power lead going to the front or rear of the vehicle.
The only other thing that can cause this is if you've intermixed the front and rear bulbs. As in using an 1157 on one end and a 2057 on the other. They look alike but the differences in the current draw and wattage is enough to cause the same symptoms you are describing.

Using a test light or a multi-meter set to read D.C.VOLTAGE ground one probe lead and turn on your lights.
now check each of the terminals in the socket to see if there is power to either of them. The brown lead SHOULD light up your testlight and or multimeter. If it don't trace the lead and repair it.the blue lead will only have power to it when the turn signal is activated