Parity is an Ethereum client, written from the ground-up for correctness-verifiability, modularisation, low-footprint and high-performance. To this end it utilises the Rust language, a hybrid imperative/OO/functional language with an emphasis on efficiency. It is professionally developed by Ethcore; we aim to have all important logic 100% unit-tested, all public APIs 100% documented, all code reviewed by multiple peers and follow a pipelined 6-ish-week release cycle similar to the Rust compiler.
Proof-of-Authority consensus method is a replacement for Proof-of-Work, that can be used in private chain configurations.
It does not depend on nodes solving arbitrarily difficult mathematical problems, but instead uses a hard-configured set of "authorities" - nodes that are explicitly allowed to create new blocks and secure the blockchain. As a result, the block creation process is much quicker than with public blockchains like Bitcoin or Ethereum - the system allows for sub-second block times. Apart from that, the PoA blockchain is irreversible and, therefore, offers near-instant transaction finality.