I don't know what complications there might be in your scene, but in a simple scene I just made this works:
If the child is sharing the same location, rotation and scale as the parent:
go to the start of the animation, select the child and press I to make a keyframe on whatever channels you want to transfer.
Go to the Graph Editor and select the Parent. All its keyframes are probably selected already (if not, press Alt A). Press Control C to copy the keyframes.
Select the child and you should see it in the Graph Editor because it has a single keyframe. Press Control V.
Now you can go to the viewport and press Alt P, and unparent the child.
If there was an offset - a different rotation or location for example - of the child compared to the parent, you could make a note of the offset at the beginning of the animation and before deleting the parent and correct all the keyframes accordingly after doing the above. For example, the child is on X -1 and the parent is on X 0. The difference is minus one. Go the graph editor, select just the X location channel at the left, select all the keys of that channel, then press G, X, -1 to Grab/move the keys one unit down.