What if the current draw is greater than the rated discharge of a battery?
Yes, it's perfectly acceptable to put banks of cells in parallel. You'll actually commonly see this in battery packs. If a pack has the designation "5S3P" for instance, that would mean that it has stacks of 5 cells in series, and has 3 "banks" of those in parallel.

Have you worked with Lithium-Ion batteries before? If not, I'd advise extreme caution for your use-case. You're dealing with quite a lot of potential energy, and if these batteries fail it can easily become destructive and dangerous (gouts of flame or explosions). Particularly in your case, you'll need to take precautions against thermal runaway (where if one cell fails and vents/overheats, the heat can spread to the adjacent cell, and cause it to vent/overheat, causing a chain reaction which can easily destroy an entire battery pack, and everything else that happens to be in the same room as it).

You should take care to ensure that the batteries are sufficiently protected, and that they have dedicated battery monitoring circuitry keeping them safe (preventing overcharge, overdischarge, temperature monitoring, overcurrent protection, etc.). You'll need to also ensure that you are balancing the cells that are stacked in series (most BMS solutions will include some way to do this). Moreover, you must ensure that the cells are all at identical, or nearly identical states of charge when you hook them all up in parallel. If one cell is at 3.9V, and another is at 3.4V, when you plug them in to one another an extremely large current will flow between them, until they balance with one another. This current can and will destroy the cells if it is large enough.