Can one run a private proof of authority ethereum network? Similar to the kovan and rinkeby testnet.
Sure. It's possible.
Check this blog post for how it's done with geth. https://hackernoon.com/setup-your-own-private-proof-of-authority-ethereum-network-with-geth-9a0a3750cda8
For parity, see https://wiki.parity.io/Proof-of-Authority-Chains.html and for demonstration of how to run a chain yourself, refer Demo POA Tutorial