What is the oldest musical instrument played other than the human voice, but body parts are acceptable? Did Neanderthals use instruments to play music?
It's probably not possible to tell. Wood and bone can often be inherently musical when struck, so some kind of proto-claves likely existed before recorded history. Rocks may have also been used for percussion from very early on.
In 2012, the BBC reported on a finding of bone flutes as the "oldest instruments" ever discovered, at 42,000 - 43,000 years old.
Obviously if simple bones, sticks, and/or rocks were used as percussion instruments in prehistoric times, it is likely impossible to tell at this point. Unless they are carved in some way, there's no way to know how they were used. Evidence of shaped lithophones (resonant rocks) exists from as far back as 10,000 years ago.
See also: Prehistoric music (Wikpedia).
One final note: In addition to excluding the human voice, I also did not discuss hand claps, feet stomps, etc., both because it seemed like the question didn't mean to include those and because none of those constitute "instruments", as I understand the word. It seems likely that the earliest music was created simply by parts of the human body.
I've been told that civilization began in Africa. That causes me to guess that drums of the sort, like hollow logs and improvised claves, might have been the original instruments. I don't know this for a fact, but I can easily picture music as we know it, starting out as simple rhythmic patterns and native chants used in fertility rites and rain dances. That said I have to add that your guess is probably as good as mine.