Thanks for the chance to assist
Currently there is no longer an "undesirable" discharge. That has been replaced by the "general, under honorable conditions" and the "other than honorable discharge
" both of which are less favorable than an honorable discharge.
It is possible to have the characterization of the discharge reviewed and upgraded. For the Army, the agency responsible for this activity can be found here: http://arba.army.pentagon.mil/
Now, there are some hurdles you face.
1. The law says that application for review must be filed within 3 years, but that the board can extend this if for good cause. This should not stop you from applying, but its important to understand this and to address in your application. If you articulate a good reason why it has taken so long...and I submit that if your father did not know if the availability of this that may be enough....then the board will consider the application.
2. You have to demonstrate the, under all the surrounding circumstances, the characterization (undesirable) is not appropriate. Ideally this should focus on the events leading up to the discharge including the process (and if your fathers rights were violated at all). However, it can also include events after the fact...such as your fathers good conduct
If you want to apply, I recommend you seek the assistance an attorney that specializes in this area. They can be of a great help in helping to assemble your application.
Please let me know if you have more questions and if not please accept so I can get credit for my work