PS2ISO.net
PS2ISO.net is one of several forum-based sites that provides links to backup files called ISOs for Playstation 2. Free registration is required to gain access to the downloads section, but that is all. Create a free account on the forums, and you will gain access to the download links. It should be noted that there are very few direct-download links, with most of the games being hosted on file-sharing websites such as RapidShare or Megaupload. Some browsing of the forums may be needed before you find the game you are looking for, but the search function is perfect for that task. The draw of this site is that uploaders often post several games in a single thread, which is great if you're looking for multiple games. The downside is that you may end up scrolling through quite a lot of screen if you're only looking for a single game.
requestISO.com
Similar in form and function to PS2ISO, RequestISO is another forum-powered backup site. This one, however, does not limit itself to PlayStation 2 backups but covers the gamut from original PlayStation, Wii, Xbox and other consoles, as well as PC games. There is also a forum where users can request specific games be uploaded. Registration is not required to access the forums where downloads are offered, but to see the download links, you must respond to the original uploader's post, and you can only do that if you are registered. Fortunately, registration is still free and only takes a few moments and a valid email address. The uploaders on this site only post a single game per thread, which makes browsing for individual games far easier than PS2ISO.net but may take longer to find multiple games.