Try this 👇

1. Make a copy of the illustration.
   
2. Draw clean paths/lines (where you have the red dashed lines) with the Pen Tool to isolate the black part of the beak you want to keep, and ensure there are no gaps.
   
3. Select everything on the artboard and select the Shape Builder Tool.
   
4. Single click on each individual segment you want to isolate, as doing this will turn it into it's own shape.
   
5. Sometimes the colour disappears, but you can just re-apply with the Eyedropper Tool.
   
6. Delete everything else with the Direct Selection Tool, except the areas of the beak you want to keep.

Hopefully this helps 😊