A fork is a "Y" shaped chain, and the true Bitcoin goes off on one of the tines, and the new "coin" goes off on the other. There is then a destructive contest as both try to become the true Bitcoin.
There is no "Y" shape. Thats just for abstraction. There is no
one chain of blocks in the "past". Each node has its own copy of the whole blockchain.
There isn't really a "contest" on which becomes the true bitcoin. Those forks only use bitcoins name to get "more established". Thats nothing more than a theft of the name.
A coin can't "go off from another". Those forks just have the same block history as btc to a specific height.
My idea of a parallel chain is to take the existing blockchain and duplicate it up to a certain height. You then run a modified version of core using that blockchain. There is no conflict or contest with the existing Bitcoin - it is a completely different entity at this point.
What you are describing here is a 'standard' fork. Every node has its own copy of the blockchain. After a certain height they run a modified version of the BTC code.
At this point a new coin has been created by a fork and is "a completely different entity".
Take BTG for example. Miner can't switch between the chains because BTG uses a different mining algorithm. The reason because miners switch between BTC and BCH is
because BCH uses the same mining algorith. And since miners are looking for the best profit they jump between those chains to get maximum profit.