Hello,
I can't give you the code, but this is my idea how to do it:
+I suppose your relationship between Parent and Child is: 1-N.
+Then in your Child form probably has a Lookup Field of Parent field.
+So in your Lookup Field event OnChange, try to implement the REST service to retrieve the Parent record to get its name --> Fill your Child field value.
+In case you create a Child record from Parent form, in Child form OnLoad event, call LookupField.fireOnChange() method.
PS: OData link