Inspector cat says No No No No No... Edit: Apparently Code says otherwise.
Returning neutral in the wrong place is a firestarter
Edit: Apparently this is allowed per ThreePhaseEel's reference, but still, I'd be happier if neutral were upsized to accommodate both sets of loads, for the below reasons.
A lot of people start out noticing that ground and neutral go to the same place. And they start thinking of ground and neutral "as the same thing". Certainly not. Ground is a safety shield. Neutral is the normal current return: the other half of the loop. By all rights, neutrals should have breakers... but as a matter of economy, they don't, on the logic that neutral can't possibly carry more current than the hot. Note that this logic utterly depends on the neutrals returning current ONLY for their own partner hot.
Your plan is to have the neutral you pictured return current for this 100A subpanel, and also, another 100A subpanel. That sounds like 200A to me, on a wire rated for 115 amps (and breakerable to 125A). Whoopsadaisy!
There are other reasons, too, which relate to AC characteristics. Basically if you don't keep hots and neutral together, you're creating some sort of "circle" the power is running around. The middle of that circle becomes the core of a transformer, with large magnetic fields being thrown. The least of this will be that your panel will be warm and buzz loudly. But the vibration can also cause wire damage, shorts, or opens. And an open neutral to a subpanel creates a real mess!
Your idea of sourcing it out of the subpanel is the right idea, but I'd rather source all of it there. First we must detour to address a couple of serious codevio's.
You must use a type of breaker that is UL listed for your same type of panel. Same-manufacturer breakers are (usually) a safe bet. Certain specific competitor breakers are UL-Classified (functionally equivalent to Listed), but they are rare (and usually Cutler-Hammer/Eaton). The issue is the bus stabs: each stab is a slightly different shape for patent reasons, and that means the wrong breaker makes corner contact, instead of surface-on-surface contact, and that matters most on high current draw breakers. Like a 50A.
Look closely at the main panel breaker, lower left. Your pictures are excellent but I can't quite read it. Look closely. If they say CL250, nevermind - that's the UL-CLassified breaker that's fine. But I really doubt that. I gather it says BR250 / Type BR-Type C. That breaker can't be in this panel. Goodbye! Replace it with a Siemens/Murray (or an Eaton CL obviously). The correct breaker is $10-ish.
But we'll use that breaker as a navigation aid. So pretend it's still there :)
Now, the breaker above it, is an I-don't-know-what-that-is. It's old. It may be a Square D HOMeline, it may be a GE. Definitely not a Siemens/Murray, they've been doing the beveled ON/OFF thing for 40 years. Byebye.
Bus stab overload
Edit: Apparently the Siemens panel doesn't state any bus stab limits. That is unusual, but believable; some of their 150-200A breakers actually lay across 2 breaker widths (i.e. 2 poles and the 2 poles across from it, precluding any use of the opposite side breaker). So maybe they can accommodate 200A.
Now, this panel doesn't have any double-stuff breakers, which makes this easier. You see the metal tabs which the breakers clip onto: those are called bus stabs. Notice that breakers on both sides clip onto the same stab. Stabs have their limits. Most stabs are 125A.
On the bottom 2 stabs, you have a 50A and a 100A both sharing. That's concerning.
Mr. Snippy didn't give us much wire length to work with, but it looks like you have the slack to exchange two breakers: a) the one above the Eaton, and b) the one to the right of the Eaton. That'll put 90A on the bus stabs in question, and allow 125A on the stabs above it, since nothing is across from that breaker. That's not too bad.
I know I've just relentlessly picked apart your panel -- but I gotta say, compared to most panels I see, this panel is very well done, with detail and panache. Even the stuff I hate (runting off the wire length; using neutral bar for grounds) is only arguable, and was done with a competent hand. And I have a feeling the alien breakers did not come from the original installer.
On to our subpanel challenge.
Simplest/hardest: Feed from the subpanel.
Edit: Since you're using #2 Al, you must use 90A.
100A 90A breaker in the existing subpanel. Land both hots, neutral and ground there. And Bob's your uncle. The issue is landing neutral and ground since I don't see thru-lugs, but I infer that you already have a plan for that.
We don't need a 100A breaker here. The only reason to use it is to give us lugs to attach to, since this panel doesn't have thru lugs.
The downside of this is that both subpanels together are limited to 100A. But we can do something about that.
Bump this subpanel.
The #2 Cu wire in use between main and old sub is actually rated for 115A. You're allowed to "round up" to the next available breaker size. So Since the busing in the existing sub allows 125A, you can change the main panel breaker to 125A just like that. Since our rearrangement, this breaker has nothing across from it, so it can pull the full 125A from the bus stabs without worry.
Well, one nitpick, I know you'll hate this, but the existing cables do not have wire markings. It was probably shucked out of cable and the markings were on the sheath. Code sees no difference between unmarked wires and vines you hacked out of your garden. At some point think about changing this stuff to #1 Cu or 1/0 Cu THHN wires, for a reason I'll get into later. Safety ground is not a big deal since the panel chassis are connected by metal conduit, and that's a valid ground path.
Now, since you're using #2
Cu Al to the new subpanel, yes, you can breaker that at 125A also you must breaker at 90A. But still, this has these 2 panels sharing 125A.
Hardest, but bestest: Bump existing subpanel to 150A
This would require changing the existing subpanel, but this buys two very nice things for us: higher capacity, and thru lugs. Also, it solves the earlier bus-stab overload problem, because we are using a breaker specifically designed to tap 150A.
Note that in Murray, a 150A feed breaker is actually a pretty obscure beast. Don't try to buy this at Home Depot; you'll leave with the wrong thing. Ask a proper Siemens dealer.
The old mains-sub feeder wires must be replaced with #1 Cu (130A; round up) or ideally 1/0 Cu (150A honest). I don't worry about copper costs for such a short wire run, and copper does save space in the conduit. But make sure to use the antioxidant goop on the lugs!
For the new panel, we specify at least a 30-space for space reasons, 200A internal busing, and also thru lugs - which will make extending neutral and ground a cinch.
To feed to the new sub, for #2
Cu Al wire, you use a 100A 90A breaker. (it's good for 115A 90A, round up to the next available size, 125A 90A).
In principle, you can take this main-first sub feeder even further, and bump to 2/0Cu and run a 175A breaker. Or 3/0 Cu and a 200A breaker or simply subfeed lugs.
Better still: bump the new feeder wire; use the thru-lugs for hots
In this scenario, we do the above 150A scenario with a thru-lug panel, but also bump your onward wire 1 wire size so it too can be breakered 150A. As a result, it doesn't need a breaker in the "old" sub. The hots can be taken right off the "thru lugs". Easy as that! I love thru-lugs.
And all it takes is a wire size bump from #2
Cu Al to 2/0 Al.
The larger wire size requires 150A breaker protection. It has it -- back in the main panel. That's perfectly fine. When daisy-chaining subs, there's no statutory requirement for a breaker in every sub.
Cable in conduit
You mentioned you planned to run conduit to the other end of the house. If that conduit will be continuous all the way, then dump the cable - it's a bugbear of a pull. Use individual THHN wires. They will pull 10 times easier than a cable, and that will make it much easier to DIY.
If your conduit is non-flexible metal (e.g. EMT), the conduit itself is the ground path. No separate ground wire is necessary.
Somebody filled your neutral bars with grounds!
Much as I love the workmanship in these 2 panels, I do not agree with the strategy of breaking the neutral bars in half and using one bar for ground. Accessory ground bars are like 7 bucks, good grief! This panel foresaw GFCI and AFCI breakers. The neutral bars are there specifically so you can land the neutral right where the hot is landed. That will make conversion to GFCI very easy. You're not in a neutral crunch (they gave you 2 neutral lugs per breaker space, in anticipation of "double-stuff" breakers), but if you ever are, look at accessory ground bars and convert the left neutral bar back to neutral bars.