I have a PC with a wireless router, that's currently providing internet access to my parent's laptop. If I got a new desktop PC, would I have to setup a new wireless connection, or will it already detect the router?

You might need to install the drivers of the wifi card(if you have one) on the new pc. It will find the connection you made, you just select it and then fill in the wep/wap password if you set one.

It should be able to connect unless it's some weird setup. Do you have another device you can try, like a phone or a PSP/DS/Vita?

Edit: What lugum said. Although if you're buying a prebuild PC, the Wifi drivers should already be installed.

Edit 2: Will be desktop PC sit near the router? If so, and the router has Ethernet ports, you would be better off plugging it in directly and not using wifi.

What they both said. ; )

If you have the wireless password or PIN stored away somewhere then you'll need to dig that up. For those with a PIN, you may have to press a Link button on the router - depends on the specific hardware. If you're not running security on your router (no first-time login or PIN required for a connection) then it's super-highly recommended that you turn it on and assign a password.

But yeah, wired is generally preferred if you can swing it.

Whay Lugum said. Note however that desktop PCs do not always come with a wifi module; You may have to add a wifi card or a usb wifi device.

However, the distance between the router and the computer and the type of materials used in the house could limit (or even impair) the signal. If you're on the same floor as the router , there should be no issue. If your room is two floors above the router and if the floors are made of concrete and steel, you may run into problems.

In that case, CPL ( power line adapters) are an affordable solution

If you somehow have trouble connecting a new wireless device, please check if mac address filtering is enabled in your routers setup. Then you must add the mac address of your new device to the list, or disable mac filtering (less secure, vulnerable if the network is unencrypted),

But as HereForTheBeer wrote, you should use wired network whenever possible - especially if it's a stationary machine.

The way you describe it you have your desktop connected to the router via network cable.

You don't have to have your desktop connected to provide an internet connection. The router runs completely independently from your desktop. You can unplug your PC from it without any problems and connect with any wireless capable PC.

Your router stores the settings required to connect to the internet in itself.