I have purchased from networksolutions a number of domain names about 2 years ago, recently I transferred one to a hosting company, I am using the site for backup and testing and to my horror I've just found that networksolutions is using it for advertising. I protected the site with a password but that doesn't seem to stop them.

Any one with some advice how to stop this?

Interesting... Can you be more specific? How is Network Solutions using your domain for advertising?

My statement above is incorrect. I thought they were advertising through a domain name I am actually making use of when they are advertising through the ones that are "parked".

When I enter "www.mydomain.org" (domain name I am using) then I get my site as expected but when I enter "www.mydomain.org" (domains names that are "parked") there is a mention:
"Site is in construction, coming up soon." followed by a page of advertising.

I just discovered this and I find the practice disgusting. These domain names are reserved for a Not-For-Profit organization that takes particular pride in its integrity, the sites are not expected to "come up soon" and these people are damaging our perceived integrity before we start.

How can I stop this?

Hook these domains up to a server somewhere and just have your own 'coming soon' page. That will stop NS using them, tho it will probably cost you a few bucks...