Feeding 125 sub-panel from 200 amp main
Option A is necessary if your present panel is Zinsco or FPE Stablok, but we can do that in a sneaky way that isn't so impactful.  Otherwise, option C.
Option A-1 is to install a replacement panel right next to the defective panel, fit double-lugs in the original panel so you can daisy-chain the new panel off the old -- then cut over one circuit at a time at your leisure.  Once all the circuits are gone from the old panel, bypass and remove it.
Now I'll discuss why option B is out of the question.

NEC: There must be a main disconnecting switch installed at every building to disconnect all service wires, either underground or overhead, and that it be located in a readily accessible place as near as possible to the point where the wires enter the building. This disconnecting means must be arranged to cut off the entire current.

This referred to DC power; the Code edition was 1897.  Current code says the same but splits it up, all in Article 230. Nonetheless, its terse language implies several things.
Only one set of service conductors per building.

230.2 Number of Services. A building or other structure served shall be supplied by only one service unless permitted in 230.2(A) through (D).

And those exceptions call out fire pumps, generators, apartments, too-large services, and different voltages/phases.  There's some blather about 230.40, which is full of very confusing terms and you will need to keep referring back to Article 100 to clarify them  - but there's nothing there that helps you.
One shut-off switch per service, practically

230.70 General. Means shall be provided to disconnect all conductors in a building or other structure from the service-entrance conductors.
230.71 Maximum Number of Disconnects. (A) General. The service disconnecting means... shall consist of ... a combination of not more than six switches and sets of circuit breakers, mounted in a single enclosure, in a group of separate enclosures, or in or on a switchboard. There shall be not more than six sets of disconnects per service grouped in any one location.
230.72 Grouping of Disconnects. (A) General. The two to six disconnects as permitted in 230.71 shall be grouped. Each disconnect shall be marked to indicate the load served.

This is no good at all.  It means even if you can snake two sets of service conductors by the inspector by snowing him with 230.40, the panels need to be right next to each other.  The service conductors would be uselessly doubled, lay right next to each other.
This &quot;Rule of Six&quot; is a liberalization of the 1897 rules, often seen in obsolete split-bus/&quot;Rule of Six&quot; panels.  Note that if your service panel is a split-bus/Rule-of-Six, a second panel would be a seventh throw, violating the Rule of Six.  The exception &quot;6 per service&quot; doesn't work because you have only a single service.
Option C
Being out of space is an intolerable situation.  We constantly advise people to get the largest panel they can bear to get, and then some. I aim to finish projects with 50% of spaces unused. Where we see full panels, we often see twin/duplex/&quot;double-stuff&quot; breakers, which have hazards of their own, not least, they actively impede upgrading circuits to AFCI/GFCI.
So right off the bat, I recommend a subpanel right next to your &quot;main&quot; subpanel, and cutting over circuits from the main to the new sub, as convenient. This will free up spaces at your leisure and at a scale that is DIY viable.

As we've discussed, double-feeding it is utterly futile.
You could daisy-chain it off the lugs of the &quot;main&quot; subpanel, but second main breakers aren't free.
Or you could wire it as a sub-sub-panel off the &quot;main subpanel&quot;, using, say, a 100A feed breaker in the main-.  The sub-sub-panel must have an ampacity rating at least that of the feed breaker. It does not matter whether this subpanel has a &quot;main breaker&quot; or not**.

As we will always recommend, go really huge on the panel.  Space is cheap, regrets are expensive.
You can move circuits at your leisure from the main-sub-panel to the sub-sub-panel.  Right away move enough circuits to liberate two spaces for the sub-sub-panel's feed breaker.
My recommendation for the addition is a second subpanel, so liberate 2 more spaces.
Option C part 2: Service to the addition
I gather from your question that you'd rather have a panel in the addition serving loads there. Just another sub-sub-panel.
If you can accomplish this with only the addition sub-sub-panel and move enough house circuits to it, that will suffice.
Again, the addition sub-sub has its own feed breaker in the main-sub.  The wire between the two decides the size of the feed breaker (and this is picked out of the tables for branch circuits, 310.16, not for service drops).  If you're looking at #4 or larger wire, seriously consider aluminum, it works extremely well for feeders like this, and the lugs are aluminum anyway.   Again, the sub-sub-panel's rating must be larger than the feed breaker. (or equal).
I recommend two subpanels to make it simple to describe, and because being out of spaces is a major annoyance.  If you can reroute enough circuits from the main-sub to the addition sub-sub, then perhaps that is the only one you will need. Regardless, don't scrimp on spaces.  A 42-space subpanel is not excessive at all. Doing all the work to install a 12-space, and then running out again, is excessively wasteful of your time.

** However, if your main subpanel is old, and maybe worthy of replacement, it would be clever to choose a sub-sub-panel which is capable of performing all the functions of the main-sub. Then you can slowly migrate every circuit over to the sub-sub, and one day, switch the service conductors over and eliminate the old main-sub entirely.