|Dial up modem and DSL modems are completely different things, one being analogue and the other digital.|
Broadband modems generally use Ethernet, sometimes USB, but never the serial port nor parallel, in fact neither dial-up or broadband modems use parallel ports.
Dial-up always uses a serial port, whether internal or external. I have never seen a dial-up modem with a parallel port. There is nothing to be gained by using a parallel port except give the modem more work to do and you dont need Ethernet with dial-up, just an RS232 serial port.
The are many different type of point to point protocols, for instance I use a broadband modem connected to Ethernet and use PPPoATM, that's PPP over Asynchronous Transfer Mode.
Just to confuse you even more, TCP/IP which is what the Internet runs on is not a point to point protocol. But you need a point to point protocol in order to connect to your ISP. It is all done with encapsulation. Look it up on Google.