The tannins won't taste good, so they shouldn't be able to eat enough to upset the tummy. I took a bite out of a live oak acorn once, and only once! VERY bitter and drying! I dare you to try one!
If you have white oak near you, I'm jealous. Supposedly, native Americans used to have wars over who got to harvest them, because of how good they are.
For some reason, I was under the impression oak was okay for parrots. I would spray it down or soak it with something to kill any nasties that might be living on it. Soaking might also remove some of the tannins.