There are many steps in the transition that are not ironed out yet. However, there are some information available that are certainly accurate for the upcoming phases of ETH 2.0.
You can transition to the Beacon Chain which is Phase 0 (see this answer for all milestones). In that step there will be a deposit contract deployed on the legacy ETH 1.x chain. That deposit contract allows you to move your ETH tokens from ETH 1.x to ETH 2.0 by creating a validator deposit.
Once you created a validator deposit, you will be able to participate in Proof-of-Stake as validator. For that, you stop running your Parity Ethereum ETH1 node and move to an ETH2 node, e.g., Lighthouse: https://github.com/sigp/lighthouse
Notably, this is a voluntary step. The ETH1 chain will continue existing for a while and you can stick to it as long as it is around.
At some point in future after the Phase 1 sharding extension is launched for ETH2, the legacy ETH1 chain will (most likely) become a shard on ETH2. This is sometimes referred to as Phase 1.5 or the ETH1-ETH2-Merger.
How this will look like in detail is not easy to say yet and it's certainly too early to say how to upgrade your clients to be able to participate in that phase.
Note, that Parity Technologies dropped support for Ethereum in 2019. The Parity Ethereum client is now on minimal life-support by the Gnosis team known as the OpenEthereum project. I'm not aware of the OpenEthereum team working towards ETH2 compatibility.
However, the Geth team is working on ETH2 components for the Go-Ethereum client, the PegaSys team is maintaining both the ETH1 Besu and the ETH2 Teku client, and the Nethermind team is working both on the ETH1 Nethermind and the ETH2 Cortex client. Maybe, to prepare for that transition, you could investigate these clients.