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

fungus:

scswift:
These LEDs are attached to a constant current driver. So there's no worries about that happening.

He's got you there, Mike...

Well only because he is running the LEDs at less than half the rated current so there is little point in having two anyway. The fact remains putting two LEDs in parallel is just plane stupid design.

And there is some kind of reflection issue in the data lines when I have more than six modules in series, where the LEDs on the last few modules will be messed up.

Yes signal integrity problems are common with this chip as people tend to think you can string them together without considering the signals. You might be lucky but just R & Cs do not normally fix things. You need proper layout and proper buffering to the signal lines.