In terms of just the SPI communication, does the Parola code do anything special that the Arduino SPI library doesn't?
The control of the hardware is in the MD_MAX72xx library. No special things are done except the initialization is by the book according to the spec sheets, as mentioned by CrossRoads. You could download the library and see if it works better for you.
With lots of modules you should be using external power. The Arduino power supply won't be enough. Also make sure that the capacitors are as per the sample design on the spec sheet as that does make a difference.
What does you circuit for one module look like? Are you using standard module so your own circuits?