ML,
I fixed the stock of a friend's Parker using Titebond II. He had dropped it and it broke cleanly from the butt to almost the pistol grip. There was nothing to clean, clean break no oil, glued it, clamped it and put a new Parker replacement butt plate and refinished just that area. You would have to look very hard to see the glue line.
I have done a lot of woodworking and have used different types of wood glue. Have glued scrap pieces using either Elmers Wood Glue or Tite Bond II or III, and in purposely breaking wood have NEVER had one break on the glue line.
If it is broken like you said, you will have to make sure that there is no oil, either from gun oil, or oil from your hands. Clean with acetone. Dry fit pieces together and clamp to see how well it looks. If you can see a highly visible joint now, you will see it once it is glued.
Becasue the piece is irregular, clamping could be tricky, if you have any surgical tubing or wood clamps that clamp on an angle that will help you.
If joint looks good, don't remove any finish from wood, clamp and then wipe excess glue with damp cloth. If you had removed finish and then wiped excess glue it will seep into pores and fill wood and finish won't come out right.

Good luck and let us know how you make out.


David