There is a workaround for this.
For example, you want to add a shared parameter to your section tag.
1. Start creating a new family using default Section Head template.
2. Switch the family category to Window Tags.
3. Put new label to the drawing.
4. It will allow you to add custom shared parameter now. You may use any of the categories which has this option available, Door Tags for example.
5. Swich the family category back to Section Marks.
That's it. Now you have your shared parameter in the tag.
I use this method to add a parameter called "View Identifier" to my section and elevation tags. The reason for this is that there is no sheet and detail number available if section or elevation is not put on a sheet (draft views for example) hence you don't know which one of your countless sections/elevations it is unless you click on it. Yes, you can add view name to your tag but I have quite long view names which would look horrible on plan. So this "View Identifier", a shared project parameter, is just a short name I give to my views, it could be anything.
However, keep in mind that as soon as you switch the family category to something else, new built-in parameters appear and old ones disappear. And it will mess with your existing labels with built-in parameters. For example when I tested this, I switched the Section Head category to Window Tags category. My two parameters inside the label - Sheet Number & Detail Number - turned both into Assembly Name parameter. When I switched it back to Section Head category, they turned both into Detail Number parameter. So be sure to create your shared parameters first, otherwise you may need to create your built-in parameter labels again from scratch.