using 14/3 with 20amp
That's "paralleling". You can't do it.

Using multiple wires to a single load is never permitted, except in industrial applications with VERY large wires, AND with special equipment on the supply side which resolves the safety issues therein, AND with very special and picayune installation requirements.  It is totally infeasible in a residence.  

Also, you forgot all about neutral

This may be news, but neutral is also a wire.  Neutral can't carry any more current than hot. So in your scheme, your solitary #14 neutral will carry 20A, which is no-go obviously. 

There's one more thing about neutral.  You notice it doesn't have a circuit breaker. The only thing that prevents neutral from overloading is monogamy - neutrals only return current from their partner hot.  Thus, the breaker on their partner hot protects the neutral.   What if a neutral is promiscuous and returns current from two hots?  It overloads, and the breakers will do nothing to prevent this! 

But wait. I'm sure I've heard of a way to get more power out of /3!

That's correct.  That is where you put 240V across the two hots (red and black), so they are on opposite poles.  In that case you can attach 120V loads between a hot and neutral, and neutral only handles the difference in current - all other current flows from hot1 to hot2.  Hot1 can supply 15A of 120V loads, and Hot2 can supply 15A of 120V loads. **But no more than 15A of 120V, ** so this does not help this here situation at all. 

But you can also ignore the neutral, and use red and black to supply 240V at 15A to an appliance.  It has to be a 240V appliance, obviously.  That is your solution here.  Tell your supplier you need the unit in 240V, where it will only draw 10A.  That will easily be supplied by your 14/3.