We have seen this behaviors once in a while. I believe you refers to the Web Part Gallery when you stated that "the web parts are not available for use on the site". The Web Parts are added to Web Part Gallery list when "Quest Web Parts for SharePoint" Feature is activated for the top-level site.
The Quest Web Parts installer automatically activates the Feature for you during installation, however there are times activation may fail or timeout. If this happens please follow the steps below to fix it:
1. On a top-level Web site, click <strong>Site Settings</strong>.
2. Under <strong>Site Administration</strong> click <strong>Site features</strong>
3. Locate <strong>Quest Web Parts for SharePoint</strong> and activate the feature.
The Web Parts available to you based on your license type should be available in Web Parts Gallery and from the "Add a web part" dialog box.