The 7219 gets it's data via 3-wire synchronous serial connection. The PC or microcontroller is used to format the data stream into something the 7219 can understand. If you can write software which can control the parallel port of your PC (I assume you can), that will be what is used to communicate with the 7219. That is, unless you decide to go the microcontroller route which Nigel suggested. The serial (RS232) port of your PC is not used in this method.
The 7219 uses three signals: data, clock and load. I would use one of the parallel port's bits as "data", another as "clock" and a third as "load". The data is shifted to the 7219 in 16-bit packets.
Basically, assert the data line with the bit (high or low) that you want to send, toggle the clock bit and after the 16th data bit has been sent, toggle the load bit to load the data into the 7219's internal registers.
You will have to preset the state of the 7219 using some control commands (brightness, decode mode, number of displayed digits, etc) before you can send it data, but once your software is written to shift the data out from the PC in the appropriate fashion, that won't be difficult.