Just a hunch, so this may get disproven...
Conduit to the rescue!
While the upper right wires going to the 15A duplex receptacle appear to be a cable (though the coloring looks strange), the lower right wires going to the 15A duplex receptacle and the wires going to the 10-30 receptacle are all going down into one location. That looks like metal conduit.
In addition, the two hot wires to the 10-30 receptacle appear to be black. Typically (but not 100% guaranteed), wires in a cable with be black, red and white (with white used for neutral if neutral is needed, as is the case here). That also points to individual wires in conduit.
In addition, there is no ground going down the presumed conduit. Which makes me think:
- If this is metal conduit and installed properly, that provides a suitable ground path. Done.
- If this is not metal conduit, or not entirely metal conduit (e.g., transitions somewhere to PVC or other conduit types) then it does not provide a ground path. However, then a ground wire (10 AWG for a 30A circuit) can be added inside the conduit.
Note that metal boxes are used as part of the grounding system whether or not you use metal conduit. So:
- Incoming power always grounds to the box first. Meaning if you do have to add a ground wire, connect it with a grounding screw to the box.
- Most better quality 15A and 20A receptacles (and all switches) automatically get ground from the metal yoke touching the metal box. So no ground wire needed for the 15A receptacle. Or at least, once you replace the 10-30 with a 14-30 and replace the junky old 15A with a nice new better quality one, preferable with screw-to-clamp connections.
- The cable to the other receptacles should have its ground wire attached to the box, not to the receptacle in this box. That is important because if the receptacle is wired up but pulled out from the box (e.g., as in these pictures), ground is not actually connected to the cable. Which means in the unlikely (but possible) situation of leaving this stuff "hanging out" and turning the breaker back on so the other receptacle works, the ground on the other receptacle would be floating and not provide any safety.