<< Serial transmits one way at a time. Parallel can send data in both directions at the same time, hence "Parallel". >>
That's not exactly true. Serial sends one bit at a time down a pipe. Parallel has multiple pipes and can send multiple bits of data at the same time.
Parallel can be unidirectional or bidirection and can run at different speeds depending on the type of parallel interface.
Serial can also run at different speeds depending on the type of interface.
Parallel & serial are just two different ways of wiring things. They can be implemented in many different ways.
In the PC world, parallel has become synonymous with the interface used by printers, some scanners, anything that uses the 25-pin connector on the back of your computer. Serial has become synonymous with the interface used by external modems, serial mice, the 9-pin connector on the back of your computer.
There are many other things in your computer that use a serial type connection - your mouse, USB, firewire. There are many things that use a parallel type connection - your hard drive, your floppy.