Setup sharing for the printer on Windows. This should be something like: right-click on the printer in 'Control Panel -> Printers and Faxes', then select 'Sharing' and enable printer sharing from there. Assuming you're on a network with your parents alone, there will be no security risks** - so go ahead and get disable whatever is needed to make printer be shared.
Then in Linux, if you're using KDE go to the Control Centre and you should be able to set up a remotely shared printer quite easily. IIRC you can even search for shared printers on the network.. user-friendliness in action.
Ps. Welcome to LQ!!
** Theoretically only. It depends how secure the entrance to your network is (eg. the router), or how easily someone else can connect to your network (more of a risk if it's wireless).