I replaced an old sub panel in my garage. The panel is fed by three wires from the main panel in the house overhead to the garage. Do I need a separate ground wire back to the main panel? There are no other sources of connection back to the house. No water, gas, or sewer lines.
You're grandfathered as long as the original installation was permitted prior to adoption of NEC 2008.
Merely upgrading the subpanel does not void the grandfathering.
You will need local ground rods, but it's still not the best situation and was banned in 2008. If you wonder why, see John Ward's video on broken PEN (Protective Earth + Neutral), which is what that bare wire is up on the pole.
If you want to retrofit ground, your best bet is probably to obtain a single wire similar to the triplex on the pole, probably XLP/USE-2/RHH/RHW-2 type. Then furl that onto the overhead service line, and also run an appropriate ground wire down the service riser. At this point, some rearrangement will be in order. The bare carrier wire should be re-tasked to be safety ground. The new insulated wire should be tasked to be neutral.
It looks to me like this pole line was thrown up by the power company, their use of triplex is typical. When you want a subpanel feeder you really need to lean on them to give you quadplex - and if they're doing it on the sly, that may not be possible since they can explain away 100' of triplex "evaporating" but quadplex not so much, since they didn't have any 3-phase jobs today.