How to represent the optionality of a many-to-one mandatory relationship from the Parent to its two Child entity types in an ERD?
It looks to me as though child_1 and child_2 are two instances of a single entity type.  

If so, then you have a (second) relationship such that 


Every parent must have one and only one favourite child
Every child may be the favourite of zero or more parents.  


The (first) relationship being 


A parent may have zero or more children (unless, of course you have defined "parent" such that a child is a pre-requisite of being a parent ;-))
A child must have exactly two parents (under basic human biology ;-))


Your first relationship may, of course, be different depending on the context.