If the other sites allow it, you could use a PHP-file that loads the image. In the file itself you can fetch and store any stat you'd like.
If PHP (or simular language) isn't allowed (it probably isn't) you could use AWstats, because it does what you want. Alternatives are:
- Analog
- Webalizer
- W3Perl
- Visitors
- Piwik
- Obsessive Website Statistics
- Etc
You'd have to check out yourself it they support what you need.