Completely Lost: Many To Many with Serializers and Update in Django Rest Framework

I've been looking at this for several hours now and I'm not finding the solution. I'm just not getting it.

I have a parent that has many children. I've created a view that allows me to get all of the parent's children. Now I want to end that list and do PATCH to the parent with the new list of children. I understand that I need to write a custom update method, but I can't figure out how to make this work.

Here's my Child Serializer:

class ChildSerializer(serializers.HyperlinkedModelSerializer):

    class Meta:
        model = models.Child
        fields = ('id', 'url', 'name',)

Here's my Parent Serializer:

class ParentSerializer(serializers.HyperlinkedModelSerializer):
    children = ChildSerializer(many=True)

    class Meta:
        model = models.Parent
        fields = ('id', 'url', 'name', 'children',)

    def update(self, instance, validated_data):
        submitted_children = validated_data.get('children')
        if submitted_children:
            for child in submitted_children:
                child_instance = Child.objects.get(id=child.id)
                instance.children.add(child_instance)
        instance.save()
        return instance

My understanding of what needs to happen is...

1. Get the submitted children validated_data.pop('children')
2. Loop through them and add each one to the parent.children many to many
3. Save the parent model

I've probably tried a dozen different ideas here, but I can't seem to get this to work. The code above doesn't change the children_set.

Any suggestions are much welcome.

For reference, i've studied the following:

http://www.django-rest-framework.org/api-guide/serializers/#saving-instances

http://www.django-rest-framework.org/api-guide/serializers/#writable-nested-representations

http://www.django-rest-framework.org/api-guide/serializers/#validation

django rest framework many to many json write

And a bunch more but I can't remember them right now

UPDATE:

[{"id":2,"url":"http://127.0.0.1:8000/api/v1/children/2","first_name":"Tom","last_name":"Jones","date_of_birth" :"1969-03-14"}]