Using single keyswitch for various different operations
Yes, you could use three separate relays with the coils in parallel but there are better options. For starters, you could use a single relay with three contacts operated from the same coil, or two relays - one with one contact and one with two contacts.

However, the power and reset buttons on most of the PCs I've built have a common connection to ground, so you may be able to connect the two grounds from each button together and connect them to ground through a single relay contact.
Here's a typical motherboard case/system connector by way of illustration, but you'll need to check your specific model to make sure this is possible:



If the LEDs run from the 5V supply, you may even be able to combine the LED ground and the button grounds and switch the whole lot to ground with the single relay contact you already have. If the LEDs run from 12V you can still do this, but you would need a diode to prevent them being powered from the difference between the 12V and 5V supplies. You should also check that the power and reset buttons are powered up together, as it's possible that one is powered from the standby rail and one from the main rail, again you may need to use diodes to ensure that they operate separately.

Finally, rather than use relay contacts you could use MOSFET or bipolar transistors as switches to turn on the LEDs and connect the button grounds.