Parity supports standard Ethereum JSON-RPC interface for mining (eth_getWork, eth_submitWork methods) and thus compatible with any miner which implements Ethereum Proof-of-Work.
First get a Parity node up and running (either build yourself or install one of the packages; the Setup guide can help you). Next, you'll need to install your preferred miner.
Just follow the instruction on the Ethereum-Mining GitHub repository page: https://github.com/ethereum-mining/ethminer.
Follow instructions on the sgminer github repository page: https://github.com/sgminer-dev/sgminer
You'll probably want to set the destination address (where the rewards go). If you have an address already, great. If you don't, then you can make one in Parity with:
parity account new
You'll be asked for a password and be given an address. Once done, you should run parity and tell it to mine to that address when required. Supposing your address is 0037a6b811ffeb6e072da21179d11b1406371c63, then you would run:
parity --author 0037a6b811ffeb6e072da21179d11b1406371c63
Once Parity is running and synced with the network, you can start your miner of choice. For genoil, command would be:
ethminer -G
Where -G for GPU mining (opencl)
Since 1.6, Parity offers Stratum protocol support (beta). This protocol has number of advantages compared to the JSON-RPC polling, which results in better hashrate and less node strain. To run parity with Stratum on, use
parity --author 0037a6b811ffeb6e072da21179d11b1406371c63 --stratum
By default Stratum listens on local network interface (127.0.0.1) and port 8008. You can change it by providing --stratum-interface and --stratum-port arguments, respectively. For example, to use all network interfaces on the local machine and listen on port 9009:
parity --author 0037a6b811ffeb6e072da21179d11b1406371c63 --stratum --stratum-interface=0.0.0.0 --stratum-port=9009
To utilize Stratum, miner also needs to be started in Stratum mode. For genoil, it will be
ethminer -G -S 127.0.0.1:8008
Parity can push work package notifications to a set of URLs by using the option --notify-work URLS, where URLS should be a comma-delimited list of HTTP URLs. For example,
parity --author 0037a6b811ffeb6e072da21179d11b1406371c63 --notify-work https://myhttpserver.local/workpackage
Once new work package becomes available, Parity will make a HTTP POST request on the provided url. Body will contain JSON-serialized array of work package data.