A new Mac user here...
On Windows, I could connect to my parents' router by clicking a WPS button on the router and then connecting to the network within a given timeframe, even though I can't remember the password.
Is this possible on Macs?
Yes, it's possible on Macs as well as Windows.
The router is doing the control of who can get in without a passcode. Just perform the same action on the router to get it to suspend the password for the next client and let the mac be the next to join the WiFi network.
Use AirPort Utility to Add Wireless Clients... from the Base Station menu for your router...
This is most likely because you stored the password on your parent's PC. Either the router requires a password (for both) or doesn't for both.
If you can't remember your password, change it on the router (refer to the manual) and set it again on your parent's computer and on your mac (once). They both will then be able to remember it.