First, I agree with boosted1g about Wake-on-LAN, BUT that assumes you will be trying to access the remote computer from a local computer. You have not told us your set-up. Are you trying to use one computer from two locations with some sort of secondary keyboard?
Anyway, the "Parallel Connection" idea is correct. There are two things to understand that make this easy.
1. The front panel "On / Off" button is a very simple momentary-contact pushbutton, just like a doorbell button. It contacts are "open"normally, and when you push the button it merely shorts together its two contacts. When you release it, the contacts open again. That momentary short of the contacts is used to trigger the actual start-up of your mobo.
2. On your mobo inside your case there is a header (usually near bottom front of the case) that has several wires plugging into it from the front panel. These include wires to the Power and HDD indicator LED's, the Reset switch, and the On / Off switch. You want to make an additional custom connection to the two contacts for the front panel On / Off switch or Power Switch. For this purpose, polarity does not matter - all the switch does is connect the two wires together for a brief time. So, you can bare a little bit of each wire that goes to the two pins for the PWR Switch and solder new wires onto them. Those two wires are now in parallel with the two original wires. From them run your line to the remote location and simply connect any simple push-button momentary-contact switch to that. Could be a plain doorbell pushbutton.
Now, turning OFF the computer from that remote location brings a different question. You will be aware that the "normal" way to shut down is use the mouse to click on screen "buttons" to shut down from within Windows. Unless you have a mouse at the remote location, you won't be able to do that. BUT there is usually a way to use that new pushbutton. At the actual computer, there is another way to force a shut-down if your BIOS settings are set appropriately, and often they are. Try this at the computer to check. Do NOT use your mouse to shut down. Exit out of all your active applications so you don't have anything running but incomplete. Then push the front panel "On / Off" pushbutton but HOLD IT IN for at least 4 to 5 seconds. In most cases, this will cause the computer to shut off completely. If that works at the computer, it also will work at the remote location where you have just installed your extra button.
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