> The function of a null modem adapter is to cross transmitted and
> received signals, right?

Basically, yes.  There are a few signals that need to be looped back as
well to do it right (for example, DCD needs to be wired to something
which is normally always active, like DTR).  The idea is that a null
modem cable looks, to the machines it's plugged into, like a
cable-modem-line-modem-cable setup that's permanently dialed up and
ready to go.

> Are there any problems doing a serial console with parallel port
> instead with serial port?

I suspect that for your purposes the answer amounts to "yes, there are
problems, and they are insurmountable".  In more detail....

It's not clear whether you are talking about a machine using its
parallel port as its console or a machine using its parallel port
rather than its serial port to talk to another machine's serial-line
console.

The latter is, maybe, doable in theory, but it's a stretch; at the very
least you will need level shifters (parallel ports use logic signal
levels; serial ports use RS232 signal levels), and you will probably
need more - a rudimentary UART, in essence, and while in principle it
could be done in software I suspect you would find it effectively
impossible to get the timing as precise as it needs to be for usably
fast baud rates.

The former is, maybe, doable in principle, but means either giving up
on console output before the kernel gains control or using special boot
ROMs that know how to talk to the parallel port.

/~\ The ASCII				der Mouse
\ / Ribbon Campaign
 X  Against HTML	       mouse@rodents.montreal.qc.ca
/ \ Email!	     7D C8 61 52 5D E7 2D 39  4E F1 31 3E E8 B3 27 4B