Neutral and ground should only be bonded at the service equipment. This could be at the service drop, the meter, or the service disconnect (250.24(A)(1)).
Looks like the neutral is bonded in your main panel, via the bonding screw.
The second panel is then fed using four wires, to keep ground and neutral separate.
Ground and neutral are not parallel neutrals. I know it looks that way because they're bonded in the main panel. But shift into a different way of thinking about the purposes of the 2 wires. Think of the ground solely as a safety shield.
Let's try a few pairs of examples. The first is Code and the second bonds at the sub-panel also. The orange glow is on things which are "hot".
Seems awesome right? Poor old Code Man is in the dark. His power tried to return via neutral, and neutral is broke, so the power failed. Rogue Man is one happy guy and his life isn't disrupted. Ground is working great as a "backup neutral". He doesn't even know he has a problem!
Of course, ground is a thinner wire, so it might overheat, but so what? Or, what if both ground and neutral were cut?
Code Man is still in the dark and he's still gotta fix those wires. Rogue Man is dead.
In Code Man's installation, the hot went through the bulb, looking for neutral. It didn't find it, so it pulled the neutral up to 120V ( not enough power for useful work, but plenty to shock). It did the same for Rogue Man, but since he tied neutral to ground in the sub-panel, ground is now also 120V, including the service panel cover and the switch plate cover screws.
Suppose the sub-panel has its own ground rod. That doesn't help much. Earth tends to have high resistance, so the cover screws might be 103V instead of 120V.
I have the good fortune of working in EMT conduit in a steel building, which naturally forces the entire conduit system to ground. Ground is never part of the circuit in any way whatsoever. So I get to see it as intended, as a protective "shroud" around all things electrical.
Ground isn't quite yet a perfect envelope. It is in new work, but we still have a lot of old wiring out there that is not practical to outlaw entirely - such as NEMA 10 and switch-loop smart switches which poach ground as a neutral.
Why bond neutral at all?
That's a GREAT question. Not bonding ground would give you an isolated system. And that makes a lot of sense in some ways, like solving some of the problems you see above. But it has other disadvantages. I go into depth about that here.
Best Answer
It depends on the supply to that property and its earthing arrangements. It could potentially be TNC, TNC-S, TN-S or TT
If it has an earth spike like you suggested it is probably a TT system. It is important you understand the distinction and I suggest you look up those earthing arrangements mentioned above they are well documented with diagrams.
In TT systems the earth neutral must not be bonded together at any point in the installation. The Live and Neutral alone make their way back to the transformer while the earthing will make its way outside the building separately and onto an earthing spike.
NOTE: The following paragraph I have block quoted and this paragraph is a disclaimer against it as it would appear subjective and only anecdotal in the experience of this region. As another user pointed out they have experienced the opposite in regard to the TT systems being historic. It does not alter the overall answer for the user who asked the question. However, I feel it adds context to the region so have left it as is.
So no the Neutral and Earth should not be bonded to each other and secondly no your sub panel should not have its own spike it should be connected to the main earthing bar within the property that will then in turn connect to the earthing spike. This is important because you don't want multiple and different impedance paths to ground. With a TNS/TNC system the supplier provides you with the earthing arrangement for you and will guarantee a (ZE) value.