Count how many children have a reference to the parent
Second problem: the views results will only include published entities if you add the filter: Published (= True). 

It's a lot more difficult if you want unpublished node to show in view results, but not be counted. 

First problem:
The easiest way to solve this is by using the Summary option of a Contextual filter. 

List the Child nodes in your Views, then add the Contextual filter for the Entity reference field to the Parent, and set it to display the Summary (adjust Validation and other options to your needs). 

The Summary will list the Parents links, with the Children count next to them. When you click through the Parent links you will see the original Views Children list (but only for that clicked-on parent). 

Don't forget the base path if the Views Display you're working on isn't the first Page Display in your Views. 

The Summary also gives you the option to sort according to the Count :) 

Your Contextual filter options should looks something like this: 

When the filter value is NOT in the URL
Display a summary
Sort order: Descending
Sort by: Number of records
Format: List
Display record count with link


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Alternativelly you could count with the Agreggation option (under Advanced settings). You would count the children title field for example, their Entity IDs, while grouping the Parent reference field.