Copy contents from an field entity reference
Using Entityreference prepopulate and Entityreference prepopulate token

Once installed, you will need to create a link for the Child on the Parent page's template that includes tokens for the fields you want prepopulated. The answer here shows how to string multiple values together.

If it needs use a Node Reference rather than an Entity Reference, then Node Reference URL Widget and Prepopluate will do approximately the same thing.