The pawl should not hang up unless the cotter pin broke and let the pin fall out the linkage. You can replace the pawl or replace the pin with the trans in the car. You will have to remove the pan. You maybe be able to get at it without removing the valve body. It will be a very close fit. 
Here is a pic from the shop manual.
You could also have an adjustment problem. Before I would dig into the transmission I would try the easier.
WITH ENGINE SHUT OFF and a helper in the car to apply the brakes should the car try to run away:
Remove the cotter pin from the shift rod lever by the bottom of the steering column and manually pull up on the shift rod so it goes fully into the park position then try to rock the car by hand to see if the park has engaged. The early Fordomatics had a very shallow cog on the output shaft drum so did not have a lot of engagement for the pawl to hold the car in park. Many 55/56 have run away if left in park without the parking brake set. You just hear a ratcheting noise as the car rolls away.
The later transmissions like FMX or Cruiseomatic had a much heavier duty notch in the output shaft drum and also much heavier park pawl.
marv
marv