I'm going to reverse my position on this and say to go ahead, it should be fine, although your building inspector would have a cow if he saw it.
PROVIDED BOTH WIRES ARE CONNECTED TO THE SAME BREAKER (which is discouraged). Perhaps you could do it with a pigtail and wire nut.
Anyway...
Say you have a single 20A breaker.
Then the whole circuit will be limited to 20A, independent of the wire configuration, and no individual wire would get over 20A.
Now, with the data you've given.
About a 40V voltage drop at 10A (assuming 120V at source).
V=IR
And your resistance is about 4 Ohms over the circuit.
That means that each leg has a resistance of about 2 Ohms.
Now, if you double the wires on the "Hot" leg, then you halve the resistance on that leg, to give you about
1 Ohm on the hot leg, and 2 Ohms on the neutral leg.
Add it together, and your circuit resistance should be about 3 Ohms, and your voltage drop would be 30V rather than 40V, giving you a final voltage for the circuit at about 90V at 10A.
You're still running 10A on all the legs, well, actually, the Neutral leg still gets the 10A, the "hot legs" would get half that, or 5A on each one.
You just couldn't install a larger breaker than the 15A or 20A breaker.
Even if the circuit was a mile long, as long as you're feeding less than the 15A or 20A into it, you should be fine, although some of your devices might compensate for the voltage drop by sucking more power.
If you doubled the neutral too, then your resistance would be halved on both legs, and your voltage drop would only be 20V rather than 40V, giving you about 100V final voltage at 10A.
Is any of this in conduit that would be easy to pull the old wire and put in new?
Also, keep in mind that only about 2/3 of the power that you're paying for is getting to the destination. If used lightly, that wouldn't be an issue, but if used continuously, then the cost of the power loss could add up.
Anyway, it is up to you. As long as you don't oversize the breakers, you should be reasonably safe, but certainly not considered Kosher.