Yes you can do this. When you set up the second router, you can chose to have the second router do DHCP and be it's own (sub)network or you can disable the DHCP of that router and have their home be an extension of your network.
By default, routers are usually set up to do their own DHCP, meaning that they get an IP address from their WAN port, and they create a separate set of IP address for each of thier LAN ports or wireless LAN connections as needed. It keeps track of all these local (LAN) addresses and communicates on their behalf up the WAN connection.
If you set your parent's house up this way, then just pick an IP address range that is different from your own. The easiest way is to pick a different number in the third set. For example you could have your house be 192.168.0.xxx and theirs could be 192.168.1.xxx. (The xxx part is a range of numbers, like 0-50, that the DHCP software will choose from for each connected computer.)
If you want your parent's computers to have access to your computers and printer etc. then turn off DHCP on the second router. This will have the second router acting as a straight through connection. When ever a new computer is connected through their outer, it will channel the request for an IP address to the first router.
By the way, I am not sure how long the distance is, but it is also possible with some wireless routers for it to receive and relay over WIFI instead of with a hard cable. If you are going to use a hard wire, make sure it is up to code for being used outdoors.