What will happen if I drive LEDs in parallel from a TLC5947?

It may not be efficient to put two LEDs in parallel attached to a constant current source only capable of sourcing 20mA, but the consideration here is for aesthetics, not efficiency.

The point is that by putting two LEDs in parallel you have no control over the current in either LED. Now you might be happy with having no control over what happens in your circuits, but it is not the way to design anything.

TI's datasheets make no mention of any special layout

Remember a chip's data sheet is designed for grown up engineers, not a kindergarten class. If a data sheet were to go into all the stupid things you should not do with it then it would be far too long.

Also, I'm not familiar with buffering, but if by that you mean put another chip on the line, that seems like it would be way too expensive,

Do you want it to work or not?
You need buffering on the signals that go to all the chips when the driver can not cope with that number of inputs. Normally I would recommend one buffer to drive 4 chips. The buffer's inputs can be common.
Layout should be tight and short to increase signal integrate and reduce pickup and radiation.

Assuming adding buffering is too expensive,

Buffers are one of the cheapest ICs you can get. You get 6 buffers in one package for less than $0.20.