Those devices use opto-isolated inputs. To all intents and purposes they look just like an LED with a resistor in series.
The built-in resistor looks like it sets the current to "between 8mA and 15mA" and typically 10mA, though it's impossible to say what it actually is. So we'll take the worst-case scenario of 15mA.
The Arduino has an absolute maximum of 40mA, and a recommended maximum of 25mA from each IO pin, with a maximum of 200mA in total through all pins.
That means you ideally can't really have more than one TB6600 per IO pin, but you could get away with 2 per pin, certainly no more.
To run lots in parallel you would need a separate set of drivers that can "fan out" the low-current signal to either a single high current signal or a parallel set of low-current signals.
It looks like you provide 5V to one pin of each input (the anode of the opto-isolator) and sink the current through the other pin (the cathode of the opto-isolator) to ground through your IO pins. That means you could use a single large N-channel MOSFET to sink all of the cathodes in parallel and invert your IO logic in software.