If I want to use a parallel port to control this 8 channel relay instead of Arduino, do I need to make any modifications?
I'm reasonably sure that you don't want to control the relays from a parallel port, but from a PC. The parallel port is a solution, not the question.
Parallel ports are so much 1980s, none of my PCs in the last 10 years had them anymore. That means you may also have problems finding the right drivers for your PC software for them. I would suggest another route. Why wouldn't you use the standard I/O interface on PCs: USB?
This module gives you 8 general purpose I/Os which you can control from your PC. The yellow jumper selects the output voltage level: 5V or 3.3 V. A low output level will switch on the relay. Each output can sink 20 mA, but the total of 160 mA is not a problem for the USB bus voltage output, since the current comes from the relay module's power supply, not the USB bus.
On the FTDI website you can download drivers for several different operating systems, and find application examples.
Parallel port from a PC??
A PC Parallel port should be able to drive the inputs to this relay module. A typical output of one of the DB0 to DB7 lines from the PC Parallel port would be able to turn on the relay coil when the port bit is at a low level and the proper Opto-Coupler / Relay coil voltages are supplied to the relay module. You do need to make sure to also connect the GND pin of the relay module to the parallel port connector GND pins (numbers 18-25 of the DSub-25 parallel port connector).
You'll need a buffer. I highly doubt your port can source 20mA per pin.