In theory, you can connect an AVR chip to your parallel port (don't forget common GND) and use assorted software to get the bootloader onto the AVR, so that it can then be used in an Arduino.
In practice, parallel ports are increasingly rare (it has to be a REAL parallel port), sneaky to configure correctly, and hard to use under modern operating systems. I don't think I've seen an reports of people successfully getting started this way...
(OTOH, the older your computer, the more likely it is to work. Maybe.)