Increasing Amp Output of a battery
For this battery it is advised not to discharge beyond 2C


A 2C discharge rate for a 3.5 Ah battery would be 7A. So, the manufacturer is recommending that you do not draw more than 7A from a single instance of this battery. 


  From my understanding, I can increase the amount of batteries in parallel to increase the capacity, but cannot increase the available current.


This is partially correct. By placing multiple batteries in parallel, you do increase the capacity, and you CAN increase the available current. In fact, most battery packs have multiple cells both in series, to increase the available voltage, as well as in parallel, to increase the available current. 

With two of your 3.5Ah batteries in parallel, you'd have 7Ah of capacity, and your 2C discharge limit would be 14A. Two batteries in parallel should be able to handle your 12A load safely. 

PS: Just be careful when trying to recharge these batteries, especially if they are Lithium Polymer. LiPos require careful cell balancing in order to be charged safely.