How do I find the network security key on a machine running Windows Vista Home Premium?
Yes, I am in fact at my parent’s house, and yes that have in fact misplaced the key. 
Go to 192.168.1.1 or 192.168.0.1 in a web browser (whichever works and gives a password prompt). Then you need to find THAT password, but if you google “default password” and the router/modem manufacturer I’ll WAG that your parents never changed it. Look through the wireless options in there to find it.
The router may also have a button to connect. You might also be able to show characters if they are already connected with the valid password (don’t have Vista to verify).
Failing that and as a last resort, reset the router with a paperclip but you might need info from the ISP.
Try this: WirelessKeyView: Recover lost WEP/WPA key/password stored by Wireless Zero Configuration service
The info is not stored in a simple easy-to-use form in the registry, so you need a tool to extract it.
Also understand that if the key is something like “ABC123”, both the router and Windows convert that into a much longer number that’s the real key. So what comes back from a tool like that is the big long number, not the easy-to-use “ABC123” or whatever was entered in the UI. You can enter the long number in the UII and Windows will recognize you’re giving the real key, not the easy seed for a real key.
At worst [ETA: as the others said just above] you can plug a network cable into the router and reset it to factory, then assign a fresh WiFi key to it and to the client machine(s). And then even write it down! 
open a DOS box (Start - Run - type “CMD” hit enter)
At the DOS prompt, type “ipconfig”
That will tell you the IP address of the PC, and of the default gateway; the gateway which 99 times out of 100 is the router.
Open the web page of the router by typing the address as mentioned by TLH.
variously, default router passwords are user- admin, password - admin, or admin and blank, or admin and “password”.
If necessary, Google the make & model of router and “default password”.
(Nowadays, Google is your tech resource manual - just gotta ask the right questions.)
Some DSL and cable modems that are also routers, the default password is the last 6 digits of the MAC address of the router (a 12-digit hex number, made up of 0-9 and A-F)
If that does not work, you may have to reset router to factory default, but then you need the login of their DSL account if they have phone company internet…
If you get into the router, you should be able to find the wireless section and set the passphrase.
Then write the necessary info - IP address, userid, passwords - on a small piece of paper and tape it to the back of the monitor or PC (not just post-it, those fall off easily)
What exactly do you mean by “network security key”? Are you using a laptop and need to connect wirelessly to the internet? In that case you probably mean the router’s WEP or WPA2 security key, which should be written on a sticker on the back/bottom of the router itself. Should be an eight-character long mix of letters and numbers.
Was trying to connect from laptop, yes, and by that key I mean the one requested when you try to connect.
In any case, thanks for the replies. I found where my parents had written it down, but couldn’t remember where…
Just wanted to add:
If the router is an older one then you would indeed need to do the other procedures described above to reset the WEP key in the router settings (for which you’d also need to know the router’s settings’ password, a different password altogether). Older routers made you choose & set your wireless password manually. As wireless became the norm however they changed this to having the wireless password hard-wired by the router and just showing it on a sticker on the device. This made it both simpler to get connected wirelessly and more secure, as the older routers would come preset with a default wireless password that you were supposed to change and most of the time people didn’t do this.
If that machine already connected to the specific network you can just view it like this guide shows.
NO 3rd party software required… works also natively in 7, 8, 8.1 & 10