Do I need some sort of driver board?
There are quite a few solutions around to interface with parallel devices like that. Some can be purchased off-the-shelf. I described a home-made solution on my page Connecting a graphical LCD via a I2C/SPI using a 16-bit port expander.
This isn't a touch screen, but from what I can make out of your (rather hard to read) image, it looks similar in concept.
I used a MCP23017 16-bit port expander (that is an I2C part, but you can get the MCP23S17 chips which has an SPI interface).
As you can see from the photo, there are only four wires from the Arduino to the LCD board: +5V, Gnd, SDA, SCL.
When I wrote that article it looks like the MCP23017 was $AUD 2.30. A quick Google search indicates it is still selling for around $US 2.50. On eBay they seem to be selling for around one dollar (eg. $5 for 5 chips).
Depending on the complexity of the driver (on the touch-screen board) writing suitable code may be moderately easy to slightly harder. It depends on how much you are used to coding for these port expanders, and how fiddly the interface requirements are.
Since Adafruit offer a driver shield, it can presumably be done.