You can:
1) Set up a no-ip.org account for both machines, tweak some firewall settings, and use an FTP server or something to transfer files, and VNC to desktop share. This is free.
2) Get a program that does specifically this (remote assistance.) Some are free, some aren't. One is built into iCloud if you have a Mac. Some chat programs include screen sharing now too. There's a free one for Windows called TeamViewer.
3) Set up a VPN so it's like they're on your LAN, then use standard Windows File Sharing. There are other advantages to this as well, depending on who has what other network devices that might be shareable (media server, MP3 library, a "nice" printer, or even something geekier like a dedicated network security appliance.) I don't know of a free VPN server, though.
Real world transfer speeds will be limited by the internet connection upload/download. So, if you have 8/1 Down/Up cable and your parents have 1.5/0.5 Down/Up DSL, a file transfer from them to you will run at 0.5Mbps, and file transfers from you to them will be 1Mbps. That's a maximum, of course, but using FTP and remote assistance applications, I usually get speeds similar to when I'm downloading files online. (So, 85-95% of my maximum on average.)