A parent chain is the main blockchain in the network. It’s called a parent chain because stemming from it are one or more sidechains. 

A sidechain is a type of blockchain that exists parallel to the parent chain. 

"Side Chains are formed by locking value in the parent chain, and sending it to a pegged distributed ledger,”  writes Delfin, an export on the subject. 

Bitcoin benefits from this by first sending funds to a specially Bitcoin address. The funds are locked and completely immobilized. Once the locked transaction is confirmed by the Bitcoin network, a message is sent to the Side Chain. This message will say that the funds are locked and new tokens can be distributed based on the locked funds. 
 

Why is this a good thing? In short: speed.

Sidechains offer a way to outsource some of the work in transactions. The sidechain, for example, could be used to transact only within one store, one game, or one environment. Then, when the work is done and the value needs to be taken to a different environment or user, the funds are sent back to the wallet from the parent chain and the parent chain wallet can then send those locked Bitcoins back out into the world.

Imagine it like a parent-child analogy: homework is given to the parent, the parent assigns it to the child who does the homework at home and gives it back to the parent, the parent then sends it out to a teacher, counselor, etc. 

There is discussion of using side chains to help scale Bitcoin and other slower cryptocurrencies. This concept would reserve the main blockchain for large transactions while the side chains take care of small transactions involving only a few cents worth of BTC. 

In theory the parent chain / side chain concept could help make BTC more of a currency -- something that people use to buy coffee with, for example.