Hey David!
I just tried as well, and it doesn't look like Autodesk has enabled us to create a parameter link to the label value for justification, so I guess that's a no.
However, probably what I would do, is create the family, and use a visibility parameter with 2 labels in the same family, and make two separate types, one for left justified, and one for right. Essentially, this will allow you to do the same thing. Looks like the parameters have to be by type, because you can't see any extra instance parameters when you load into the project.
1) Create two label types in your family, one that is left justified, and one that is right justified.
2) Create two new parameters (by type). One Called "Left Justified" and one called "Right Justified"
3) Create two New Family Types, One Called "Left Justified" and one called "Right Justified"
See the attached File.
This is far from a perfect solution, but could work for now.