Can I combine two 12-2 cables to serve a NEMA 10-30 socket for dryer?
Other answers cover the reasons pretty well.
I fail to understand how running a second cable in parallel should be any more expensive than just running a properly sized 10/3 cable with L1, L2, N and PE conductors, to accommodate any dryer on the market, and disconnecting the original #12 circuit. The only price difference would be the cost of the cable, and that's small compared to the labor involved anyway.
The discussion below is from an abstract electrical engineering angle and completely ignores codes and similar requirements. Let's just think of what it would take to parallel the wires safely, in general, assuming a typical US residential panel as the source of power.
You'd need to protect each wire individually from overcurrents. For each paralleled wire, you'd need a circuit breaker rated for the ampacity of the wire (or less), to protect that wire from overheating.
Let's say we want to parallel #12 wires to use in a 30A circuit. You have 2 L1 and 2 L2 #12 wires (i.e. both wires live). In the electrical panel, you'd need a 20A (or less) circuit breaker for each wire - four circuit breakers total.
But I'd say that this is cutting things too close - by the time one of those 20A breakers would trip, there'd be a fairly large current imbalance. You'd want smaller breakers so that they'd trip when the imbalance is developing but far from overheating the wires. Thus a rating a bit above 15A, but less then 20A. But no such breakers are made for branch circuits in the form that would fit into typical proprietary (e.g. GE, Leviton, etc) panels.
So, we'd have to use DIN-rail mounted breakers, UL-listed for branch circuit use, and select e.g. 16A breakers. So this becomes fairly complicated now. Here's what it'd take:
2 x 30A branch circuit breakers on the L1/L2 feeds in the main panel, going over #10 or #8 wire to a subpanel (I'd prefer #8 since it's more robust when feeding common DIN parallel feed terminals).
4 x 16A branch circuit breakers on a DIN rail in a DIN-rail subpanel. The circuit breakers feeding each pair of wires would need to have a common trip mechanism, so that if one of the wires in the pair trips the breaker, the other wire would be automatically disconnected as well, since there's no way it could do its job safely then.
In the receptacle box fed from these paralleled #12 wires, each pair of wires would need to be connected to a #10 pigtail, and only that pigtail would connect to the receptacle. 30A receptacles only allow connecting a single feed wire, so the pigtail is necessary.
And when putting this into service, you'd have to use additional wire lengths somewhere to do equalization loops so that the "cold" current flow in both paralleled circuits would be equal. Obviously the wire lengths would have to be quite similar, and ideally both cables would run in the same thermal environment, e.g. next to each other (at which point you can just put a proper cable there!).
The above is not legal in US jurisdictions, but from an engineering standpoint it is a reasonable approach towards a solution assuming that nobody has an urge to tweak it later. And this assumption is laughably wrong. So, it's the sort of a "reasonable" solution that is only reasonable in a lab setting, and thoroughly impractical anywhere else, because humans are not perfect.
Let's recap: instead of just running the properly sized cable, we'd have to install a whole new subpanel with its own breakers, and then ideally run another cable right next to the original one. That doesn't make any sense anymore.
Now one could argue loudly that it's "unfair" that the NEC doesn't allow such kludges: there's a good reason for that. A kludge of that sort would have a very low probability of being installed right, and also would likely be installed in circumstances where the original cable run may not be up to par (e.g. there may be hidden splices or who knows what in the walls). And there would be DIY types who would be installing such kludges left right and center. And then whenever the 16A breakers would trip due to the inevitable imbalances as the old connections oxidize and so on, there'd be a tendency to use larger breakers instead or to bypass them altogether since they'd be "a nuisance".
Sometimes, the codes prevent theoretically reasonable engineering measures because there's no way to trust the humans executing the kludge to get it right, and then the humans using the kludge not to make things unsafe. This is also borne out of death and ashes. The codes have it totally right.
This sort of a kludge could probably be put in place on, say, the International Space Station, since there'd be enough oversight and understanding to mitigate any developing problem. But that's entirely hypothetical anyway, since none of the usual parts used terrestrially have any place on ISS due to offgassing, toxic fumes when overheated or when on fire, inability to operate in vacuum (suppose there's a decompression - last thing you want is for a power distribution unit to catch on fire because there isn't enough convection cooling), and so on. I'm just talking of the "class" of kludges, not about literally doing it using the same parts or even when solving the same problem.