There might be a better way of doing this, but I’ve found success with this method.
Instead of just parenting the object to the hand bone, position the object where you want in the characters hand and snap your cursor to the selection. Then create a “child of” constraint on the object and make it the child of the bone. The object might translate off in some weird way, just snap the selection back to the cursor. Now you will be able to animate the influence of the parent-child relationship. keyframe the influence from 0 to 1 over the course of 1 frame to pick it up and vice-versa to put it down. Keep in mind that when you put it down, the object will return to its original location before it became the child of the bone. To fix that, use the same method mentioned above by snapping the cursor to the object before you turn off the influence of the constraint. Then after you keyframed the influence off, snap selection to cursor (be sure to keyframe the location of the object 1 frame before it is put down, then snap to cursor and keyframe it again).
Another solution to your problem would be to render out the frames where he is holding the object, then delete the parent and seperately render out the frames where he isn’t holding it.