I had this happen once - car would not stay in reverse when OD was engaged. Turned out a snap ring inside the trans jumped out of its groove. That's likely not the cause as that was very unusual. According to the shop manual, if the spring inside the OD rail/lever in the trans breaks, it can cause this symptom.
And yes, if you have the OD control pushed in, you must park in reverse.
Sounds like you do have an issue. Without opening the trans, it's hard to say for sure. It could be the rail switch spring.
The other time I saw this symptom was on a '51 Frazer where the owner installed a backup light switch which was operated by the lever on the side of the trans. That switch got jostled out of place and would not allow the transmission to go into reverse. A visual inspection of the outside might reveal the cause. Have a look first and see if anything might be obstructing the movable parts on the outside of the trans. Here's hoping it's something simple like that.
