Up until 2011 the National Electrical Code allowed a "3 wire" method of connecting a sub-panel. This means you have the two "hot" wires and a neutral running to the sub-panel as opposed to the "4 wire" method where you add a separate equipment ground wire.

So, when your house was built, it was perfectly legal to use the 3 wire method. There is no problem with this unless you connect a metal pathway, such as a water pipe, from one building to the other. This allows neutral current to flow on this parallel path. This could be potentially lethal to someone working on the pipe.

As to the main panel not having a neutral: it does. If you look again you will see the two hot wires and a bare conductor. There is no ground wire on the service drop. The bare conductor is the neutral. This is allowed on the service drop, but from that point on all neutral conductors must be insulated. Ground wires are always allowed to be bare.

So, in short, there is no problem with your system unless you run a metallic system to the sub-panel building. This would create a problem.

Your grounding problem is due to the open N-G bond in the service disconnect

An outlet tester's "open ground" indication is an indication of an open somewhere between the ground contact on the receptacle and the N-G bond in the service equipment, as the light in the tester that corresponds to this is connected from Hot to Ground, and if the bond is intact, that circuit will be completed, even if the grounding electrode system is missing altogether. However, if that Neutral-Ground bond's gone, that'd explain the open ground reading you see.

As a result of that, I suspect the grounding electrode system to the disconnect is not the source of your issue; note that the pool tech was correct in disconnecting the redundant GEC to the pool pump as the pool pump and pool bonding grid are bonded to the mains equipment grounding system through the pool pump circuit's equipment grounding conductor (if nothing else). The grounding electrode should still be examined while you have qualified eyeballs in there, though.

REPLACE THAT PANEL!!!!!!! (and the service disconnect too!!)

The busted RMC for the service is actually not the largest safety concern with your parents' electrical installation, though! That subpanel was made by Federal Pacific Electric, and is an outright fire hazard! Many of those breakers cannot be relied upon to trip if a circuit is overloaded (either due to common-trip jams or simple miscalibration), and the Stab-Lok design is prone to contact faults in the breaker-to-busbar connections as well, which can lead to fires within the panel itself. As a result, the only fix is to replace the entire panel with a new one.

So, the verdict is:

1. Replace the FPE panel with something decent, see below for replacement suggestions
2. Replace the disconnect with something decent since it's also a FPE atop having its guts rusting out on you -- note that it can be a different flavor than the subpanel
3. Have the damaged RMC service conduit permanently repaired or replaced

Panel replacement options

There are two criteria I use when choosing replacement panel types:

1. It needs to be available in >42 slots (42 is a bare minimum slot count for a main panel)
2. It needs to be available in both single and three phase version (this is a sign of a commercial grade panel, not something that's purely builder/residential grade)

This leaves you with a few options, depending on what you can get (i.e. some supply houses are partisans to one manufacturer or another) and whether you can rip out the existing enclosure:

• Square D QO and Eaton CH are both considered "industrial grade" lines, but they are slightly costlier due to using a 3/4" form factor. They do allow you to get a better slot-to-space ratio, which is a factor if you're using an Eaton retrofit kit in the existing enclosure.
• Eaton BR is a midrange line (some would say it's builder-grade with industrial aspirations). It has readily available exotics and a good variety of options, while being inexpensive and widely distributed.
• Siemens (QP) is one of the few "industrial grade" 1" breaker lines out there. It is less costly than a 3/4" form factor panel, while providing basically the same level of performance, and access to >42 slots.
• Last but not least, if you can't rip out the existing panel enclosure at all, Eaton has retrofit interiors available in both the BR and CH product lines that can be used in this application. You're limited on slot count by the existing enclosure, though, so this is kind of a last resort compared to putting in a new panel that will let you take advantage of the code changes over the past 15 years that liberalized panel sizing.

As to the service disconnect, it can be replaced with a type of device called an "enclosed circuit breaker" such as an Eaton ECC series if you reroute the swimming pool circuit to the new subpanel, or with a standard 200A loadcenter of some sort -- in fact, this'd be a perfect application for one of the small feed-through-lug equipped loadcenters Eaton makes, such as a BR816B200RF. That way, you don't need to burn 4 slots on a 200A subfeed lug in the outdoor panel, while the swimming pool circuit can stay routed the way it is. In addition, you may want to have the conduit rerouted so its not trying to drain water that gets in straight into the panel, or at least ensure that the new install has the top conduit fitted correctly so it isn't leaking water into the panel.