It is a simple enough matter to "bit bang" a pair of the IO pins of a parallel port to emulate a serial port. It's all in the software.
And that's where it all gets a bit tricky.
The basic operations of reading and writing are simple enough - what is hard, when you are working on a PC with an OS in the way, is the precise timing needed to read and write at the right times.
Programming "software serial" on something like an Arduino is pretty simple because you have hardware timers, and direct access to the hardware.
Doing it on a PC when you have so much more going on would be a rather tricky task. And then of course there's the writing of a driver so it's seen as a COM port, etc.
So although it would be possible, it's really not worth the huge amount of effort, when all you really need is a little USB dongle that you can pick up for $3 on eBay.