I had this same issue. I started a new project with only a previously configured Parcel Fabric layer, but this was still unable to 'grab' parcel points. Before submitting a case to ESRI I started another new project, but instead of adding the old Parcel Fabric layer, I added the raw Parcel Fabric data and it allowed me 'grab' parcel points.
Please contact technical support to create a case for any unexpected behavior.
Cases that are software bugs are reported to the development team.
Thanks.
I didn't open a support case with ESRI as re-adding the Parcel Fabric from source -- instead of using a layer file or opening another similar project -- allowed me to move parcel points. If this issue later becomes a problem I will open a case.
The original thread is from 2013, but it is hard to tell if it is the same or not because you might be using a different version of the software, different DBMS and many other variables could be different.
Please open a case and we'll have a look. Your insight about the layer is very valuable.
Thanks.