Why can't large batteries (such as the one by tesla) be charged in parallel
Not only is @supercat's comment correct about the charge cable limit, but each individual cell limits the charge speed. The "supercharger" speed is 40 minutes to 80% (145kW!). Which is about as fast as it's possible to charge a single 18650 cell without it overheating.

The limits within a cell are due to internal resistance (which causes self-heating) and the diffusion rate through the electrolyte. Most of the research on fast charging cells focuses on building electrodes with a "fractal" extremely large surface area but also acceptable durability.