I have organized the customer hieracrchy as follows:

1. Customer
2. Job
3. Task

A job has multiple Tasks that are unique to the customer, and may repeat Job to Job.  Once a Job is complete, we move all Tasks in that Job to a Job for that customer called Complete.  When a new Job is added for that customer, our first action is to move repeating tasks from Complete to the new Job.

The Table Detail Reference tell me ParentListRefID and corresponding fields cannot be modified.  How can I use an Access routine to move these Tasks among Jobs.

As the customer (ParentListRefID) is always the same, you would normally change (update) the transactions from one customer:job CustomerRefListID to the other customer:job CustomerRefListID

Each job or task has its own CustomerRefListID. See: Adding Customer - Job 

Using the VBDemo utility, I was able to successfully move a task at sublevel 2 for a customer from one job at sublevel 1 to a different job at sublevel 1, using the ParentRefFullName field.

The QB data base immediately reflected the change...the VBDemo query executed immediately after the update did not.  I understand this to be a Sync issue, so I will have to do a resync after executing this command, which may be many times in one day.  What sync option do you recommend rather than resync'g the entire file each time I execute my routine?

See: How do I setup the QODBC Optimizer? Where are the Optimizer options?  My recommended settings for using QODBC are: