Well, there is a reasonably easy way to do it that mostly gets it right, and a considerably more complicated way to get in most of the way to right.
As MonkeyHead said, you need to create your stockpiles and get down into their settings with 'q' and then 's':
In the Refuse section, you can select body parts and items. However, there is an issue in that remains of Forgotten Beasts and Titans are unlisted, so the only way to exclude parts of those from a stockpile is to go down into the Refuse submenu and select 'b'lock All and then add what you want. For instance, I make a stockpile that only takes Fresh Raw Hides in my tanner's workshop, and then exclude those from the other piles by going down into the Item Type sub sub menu.
In the Refuse submenu you'll find the cateogries you probably want, and you might want to leave it at modifying those.
However, body parts from (sort of) sentient creatures spook dwarves and can get them into despair if regularly exposed, and caravans can likewise be spooked by remains, toss the stuff the spooked caravan member carried on the ground, and the caravan then leaves without a notice whatsoever (suddenly you don't have any "haul to depot" jobs and you can't order any, and the caravan has reversed direction).
For that reason, you will want to dispose of sentient parts separately, and that can be done in two ways:
1. Locate every "sentient" body part and Dump it at a suitable location (such as a magma pit). This will interfere with with usage of dumping for other purposes, however.
2. Create a separate "sentient" refuse pile at a place off the side where it can't be seen (I typically have a wall with a door in between). Unfortunately, the process of doing this is very labor intensive, because you will have to manually go into each of the sub sub menus to add/exclude every sentient creature species individually. I typically start with goblins, trolls, and ogres (blind and normal ones), and then fill the list up as I encounter new types (Plump Helmet Men, Reachers, Gorlaks, beast men...). Note that this will have to be done on EACH of your stockpiles (except the tanner one, since Fresh Raw Hides is an item type).
If I knew how, I'd like to define the lists in a file once and then use DFHack to apply those definitions to my refuse piles, similar to how copystock works.