Maybe you could use another system to do some kind of signal conversion. Maybe a Raspberry Pi fitted with a serial to usb adapter could get data from that serial port and send it to a network printer, but I don't know if there are software that do the trick.

If you want to use parallel port, I guess the best option would be make a parallel to GPIO adapter.

To make thing easier, your printer should be able to understand a PDL supported by your application (i.e.: PCL, PS)... but also you could get the data, convert it to another PDL (using ghostscript?) and send it to your printer.

(I've got a Raspberry Pi connected to a USB only printer, to use it to a network printer)

Another option would be capturing the LPT output... there were some programs that could capture all LPT output to a file, or (if you have plenty of RAM) using a network client and connect LPT2 directly to a network printer.

pretty sure you share a printer using an earlier version of windows like 9x or nt,

then install msclient on the dos computer and map the shared printer using net use.
net logon
net use \\PC_name\Printer_name

That will map the LP1 to your network print queue.

Then once you have it mapped, you can use command like
print filename
or
copy filename lpt1
for example copy c:\hello.txt LPT1

You might be able to do something similar or even better with netware with lan workplace for dos, although I never tried it. you would have to use novell tcp ip stack.
http://josh.com/tiny/lwp/dosutil4.htm

Or use a TCP/IP stack that offers such printer mapping to LPR, to send the stuff to the printer directly.