You are on the right track. But to be certain you should go to the actual light socket on one of the running lights that isn't working. Remove the socket assembly and remove the bulb. Check to see if its clean and free of corrosion and rust..
Use a "test light probe" like this. link to test light pic and general use instructions
When you have the light bulb out you can do the test either with the connections inside the socket or you can use the probe on the back side of the socket.
First you want to check to see if you have power going to the light to start with.
1. connect the ground connection to something you are certain is attached to the frame, and has no paint. Screws, bolts or bare metal are all good. Then with the parking lights on
check all of the posts in the socket you should get a light on one of them.
If you don't double check your ground on your test light and try a couple different places and a different power source. You want to KNOW you have a good ground first
If you do get a light on the test light, remember which socket you got power from. If there is only one other socket it will be ground.
Some cars have a dual power source for parking, brake, signals etc.
Next you can check the ground by putting the test light connection in the socket you are sure has power. and the probe in the other socket. you should have light on your test handle. If you don't leave the connector where it is and stick the probe on something metal to see if that works. If it does then you do have a problem with the ground.
BTW having a bad ground is not very common.