Is it possible for my parents to spy on my internet... from afar...?

Yes, it is very possible, there are ways to protect yourself though. Clear those things first off, you don't need those chat logs, you don't need a ton of history of pages, you don't need a cache filled with endless pron sites, err, or whatever else you visit. As well, make sure you use pwords for your Account to log into the computer, the bios, and if at all possible, to get it to allow you to open up the Internet. Another trick, when not on the Internet, remove yourself from the network, or, if you can access the Internet while not on the network, then don't be on the network when you don't have to. If you know anything about setting up firewalls, you can do that to prevent it from opening the port that they need to access to obtain information from your computer, or at the very least, at least prevent it from downloading/uploading anything from that port that you haven't already given permission from. However, if you do all this your parents may begin to wonder...