Can't you just do a run-as admin and ask them to provide the credentials? This would install the software and I assume you will be able to run it as a non-admin afterwards.

Linux =/= Windows. Linux is designed around a multi-user environment and expects you to install applications within your individual account without necessarily modifying anything in the core system directories. Even with that, Linux still follows the standard practice of requiring admin permission to install anything that will be globally accessible or anything that requires a modification of the core system directories. Windows, while definitely more multi-user oriented since the switch to the NT core in XP, is still following the single-user philosophy and expects you to install most software for all users at once. While this is monumentally stupid, they at least have made so that you do require admin permissions to do it, with the expectation that an admin knows what they are doing (almost never the case). Even when it gives you the option of installing for a single user, all it really does is install exactly where it would install everything else (i.e. core system directories like Program Files) and only place shortcuts on the installing user's desktop and menu. Then there is the ever-present problem of the Windows registry, something that doesn't even exist in Linux. If an application does not modify the registry at all and it gave you the option of installing somewhere writable, like a "My Documents" directory, then a guest account would probably have no issues installing. However, none of the GOG games (that I am aware of) do that. All of them have some interaction with the registry that is required for the normal operation of the game (DOSBox and ScummVM games may be an exception to this). There is nothing GOG can do to change this as it is part of the game, not the installer.
The only real solutions to this problem have already been presented: either have his parents give him his own account or use the "Run as" option and let his parents provide their credentials for install.

Adding another option/opinion on the bucket.
Google chrome can be successfully installed by a guest account (just tried it just to be sure) because it installs to the account, not globally to the machine. At first run it asks for admin credential for something (some registry changes, I suppose), but denying that, it still runs.
I don't know how they do it, but it proves the installer could be written to work that way.
Of course, that means every account wanting the program/game will need their own install, multiplying the amount of space needed.

Yea, like I said this isn't unheard of. Without registry access you just use an .ini file for settings.
Not even sure what, especially old games need with the registry anyway. Maybe uninstall information, but even then you could store that in a local file.
BTW Gnome does have a registry.

Not in the sense and the scope of Windows' registry AFAIK.
As for admin credentials, I find it logical to be necessary: a normal user shouldn't be allowed to install software in any circumstances as that provides a vector of attack for malware.

Doesn't really matter when most regular users can run scripts in a word processor / browser that are just as harmful. Basically any damage the user him/herself can do the malware would be able to do. But it wouldn't be able to bring down the entire system.

Your parents need a good hacking.

Wouldn't it technically be better practice for them to NOT require administrator access?
Really, only a few programs should require admin access, things like virus scanners and defraggers that need to access restricted resources. The reason all software needed admin access when vista launched was that it hadn't been written properly. All the UAC popups were actually MS's attempt to get people to write better, safer, more contained code.
(which was admirable, and has actually worked - but unfortunately annoyed all the users in the process).
That said, because most GOG games weren't written with vista in mind, installing them without admin access can often lead to errors and compatibility problems. One of the first thing people always suggest when you can't get an old game to work is to install as admin.
So making all games install as admin is probably an extra step to prevent compatibility problems.

I suspect many GoG installers need admin access because the games themselves require admin access to install. A lot of the non-DosBox games at least stick a few values in the registry. There's no way around that sort of thing without modifying them source code of the game or running it in some sort of jail.

Gog should let you choose the install path. And you could install to a location that normal users have write access to.
So theoretically you could install all your games to My Documents if you chose to.
Sure a bad user could fill up the hard drive that way, but a good admin would put quotas on the user's home directory.

GOG does let you choose the install path (doesn't anyone click on the "Options" button in the installer?). The install path is most likely not the only problem though, there is still the registry. It doesn't matter where you physically install the game, it will still need to write registry values that only a user with admin permissions is allowed to do.

Haha, I've never used that button. :P
But yeah, that's why the registry sucks. It's so much easier for apps to use configuration files (.ini or .conf) instead of dicking around in the registry.

Most games, even the ones from GOG, do use an .ini or .conf file to store settings, that's not what they use the registry for. Its used to register files with the OS, set application paths, uninstall strings, etc., most of which cannot be done on Windows with an .ini or .conf type file. Its a flaw in the way Windows is designed that has existed since the introduction of the registry way back in the day.
EDIT - Hey, something has changed with quote system, it didn't screw up on me with three nested quotes this time.