Can an iPad (3rd gen) run solo with an Airport Express
Edited since I misread the question

You can definitely set up and control any Apple router that supports 802.11n from an iPad only. Apple has an iOS app just for that purpose.

However if you want to save some money, or possible use the existing router, that may be an option too. Almost all routers these days have web interfaces, so an iPad can manage most routers. However there are some routers that require a PC app to manage, typically older ones.

Chances are they'll be fine, but you'll first want to check what kind of router it is, then have a quick look at the manufacturer's page to figure out if it has a web interface or not. If it does, you'll probably want to figure out the IP (typically 192.168.1.1 or 192.168.0.1) and bookmark that for your parents.

It is a bit troubling that the router crashes when the PC crashes - sounds like something's not quite right there, so definitely check that out.