Your panel was wired improperly in the first place. But we can save it.
What they did was combine two cables to get enough wires to do the job. That's not allowed. I would call it "cheap" but they actually spent a fortune compared to what they could have spent.
So let's fix it on the cheap, using the existing wires.
You will need two subpanels. You already have one, so you'll need one more. One panel can be 120V and the other one 240V-only (no neutral). Or both 120V. Or both 240V-only. Your call.
Each cable goes to a different subpanel.
The wires are only good to 55A and you can only plan to put 55A of load on them. However, since 55A breakers are not sold, you can use a 60A breaker. This is not permission to plan to use 60A of ampacity.
To wire a 120V-only panel
On the 120V panel(s), the white wire goes to neutral both in the subpanel and on the main panel. On the main panel, the black wire lands on the breaker - the other half of the breaker is empty.*
At the subpanel, the hot wire is split to feed both hot lugs on the panel. Lugs are only rated for 1 wire, so both lugs will need to be pigtailed and tied with a 3-wire splice. 120V loads will now work. Do not use MWBCs.
A 120V panel can support 6600 VA of calculated load. (120V x 55A).
To wire a 240V-only panel
The white wire at both ends gets taped with black or red tape to indicate its use as a hot. It goes to both poles of the 240V circuit breaker, and to the two hot terminals on the subpanel. At the subpanel, keep the neutral bar completely empty (this panel does not have neutral) and buy an accessory ground bar - this is to make clear to "the next guy" that there is no neutral here.
It'll need to be a nice big panel, because you'll be using spaces 2 at a time. Spaces are cheap.
You then wire up 240V-only (no neutral) loads, such as a heater, air conditioner, water heater, welder, compressor, or EVSE that is hardwired or uses a NEMA 6 family socket. You cannot install a NEMA 14 socket, but EV's don't need neutral.
If you have an EVSE with a NEMA 14-50 plug, either send it back, or throw away the plug and install a NEMA 6-50 plug (notice when disassembling the 14-50 how there is no neutral wire in the cord, and the neutral pin connects to nothing?) EV's have no use whatsoever for neutral, so the 14-50 socket is pointless and stupid. I have no idea why EV suppliers love that plug so much.
A 240V panel can support 13200 VA of load. (55A x 240V).
Two of them can support 26400 VA of load, more than the 21600 VA you were asking for.
"They are 4-wire 120/240V loads, or I can't say."
Then you have no choice but to replace the entire cable run.
I would use 2-2-2-4 aluminum wire of a type such as SER which is rated for the same places NM cable is rated. There is nothing wrong with aluminum at these large sizes, and the panel lugs are aluminum anyway.
Now here's the kicker. Price 2-2-2-4. Now price 6/2 copper NM. See how the 2-2-2-4 is about 1/5 the price of what they did put in? This is the trouble with using wire "you have laying around". They could have sold the 6/2 on Craiglist for more than the 2-2-2-4 cost, and had plenty left over for a larger subpanel.
2-2-2-4 aluminum is good for 90A. Some people believe it is 100A wire - there's a reason they think that, but it's wrong for your application.
This will support 21600 VA.
* They don't make 1-pole breakers larger than 30A, so if you need a 120V breaker that is 50A or 60A, you will need to buy a 2-pole breaker and use only one pole. If your setup is dual 120V subpanels, you can use the same breaker (but they'll trip together if you do).
