There's plenty of information on the net about booting from USB. Infact, alot of Linux systems can boot from USB (such as DSL, and Knoppix).
Get yourself a normal HDD, maybe a 500GB one, and then convert it to an external HDD with the minimal setup fee of simply buying a case and converter.
Once you've done all that, you have to set your parent's PC's BIOS to boot from USB, so you can plug your USB drive in and boot from that.
Forget about any kind of fast loading since you have to run it through USB.
As for their registry "corrupted to the point where it can't be fixed even in safe mode or by reinstalling windows", any reinstall of windows will fix the registry.
Don't try and say it's a be-all and end-all, because it's not. Your HDD is your HDD, no matter what MS tries to tell you, you can format it and reinstall windows as you like, and each time you do, it will write this little section of data to your HDD which just so happens to be called a "Registry".