Mac email has strange numeric charset encoding problems?

What you’re seeing is a problem with what are called character encoding on the computer. Whether you’re using a Mac, a PC or even a Linux system, every message you type is encoded using one of the dozens of different language and locale options available on the system. If your message includes the information about that locale or character encoding, and the receiving system has the same encoding system, everything’s magic, whether it’s Farsi, Cantonese, Hebrew or Russian.
If there isn’t a match or the message isn’t being properly identified when it’s being sent, however, you can get all sorts of strange gobbledygook that can leave you trying to puzzle out exactly what’s going on, and that’s what I bet is happening with your administrator.
How you set the character encoding is different in different mail programs, so let me show you how I do it in Microsoft Entourage as an example.
There are two different places you can tweak this, actually. On the outbound messages the program should automatically pick up the character encoding and locate you’ve specified for the computer itself, but as you’re sending a message, you can also specify the character encoding with Format –> Character Set, as shown here:

Just about always “Automatic” should work fine, but you can also pick a specific encoding if, for example, you use your computer in English but want to send a message to your Aunt in Japanese. For regular use, “Western European (ISO)” is probably your best choice.
If you want to see where you character encoding and locale are selected on your computer, rather than within Entourage, then you should go to Apple –> System Preferences… and choose “International”. Then in “Input Menu” you can see what you’ve selected:

Back to Entourage, though. If you’re finding that just a few parents are complaining about the weird embedded characters, it might well be that their systems are set up incorrectly! They should check the setting Entourage –> Preferences… –> “Read”:

As you can see, the key setting is “Default character set for unlabeled messages”: if you have that set to something like Korean, the majority of your messages will in fact be weirdly unreadable. Set it to something like Western European (Windows) and you might just see the occasional odd character or character sequence show up.
If the administrator is running Apple Mail or another email program rather than Entourage, you’ll find that the settings and preferences are quite similar, albeit not exactly the same. Nonetheless, the key setting to dig around and identify is character encoding or just encoding and hopefully you’ll be able to fix this mysterious problem!