I am pretty sure they should not be bonded together please correct me if I am wrong. Now the feed to the subpanel only has three wires so how can I tie in a ground? Run a new wire to the main panel? Is there any other way it can be done?

I added a picture of the panel

The panel is in the besement across the room from the main panel. Attached structure

 

The installation may have been code compliant when installed but there is no way to bring it up to current code without replacing the cable because it doesn't have an insulated neutral and you can't run a neutral that is outside the cable sheath.

 

Looks like you only have four circuits in the panel. If there is room in your main panel across the basement, you could extend those circuits to that panel and remove this panel and feeder. If you have plans to add loads then I think you need to install a new feeder. The panel can is not grounded as the strap is not connected to neutral with all the other grounds, even though they should not be combined. Correction is needed.

 

Aside from not being code, Is this a safety issue? There is room in the other panel however this panel is in a great location and I planned on adding some things to it.

 

It is a code violation as shown and needs to be fixed. A ground bar will need to be installed after the 4 wire feeder is run.

 

Yes. That's why the code was changed.

That panel is essentially on an ungrounded circuit. It cannot be extended and be code compliant. You need to either replace the feed or abandon the panel.

 

OK thanks. I have some service cable at my dad's house. The next time I am there I'll grab it

 

That panel as it is presently wired may be used for 240 volt only circuits and appliances and equipment. The bare conductor would be a legitimate equipment grounding conductor.