Is the “Delete this record also” option for Lookup relationships available only after getting the ‘Cascade-Delete’ option enabled for your org

The relationship fields on Activity aren't actually Lookups, but polymorphic relationship fields. This is a special field type that the platform uses when a child record can have a relationship to more than one type of Parent. They aren't available as a field type to Salesforce users when defining new relationships in the database but they do show up on a few Standard objects.

They behave more like Master/Detail relationship fields when it comes to parent deletion.

If its a lookup you will need a simple trigger to do this in real time

trigger onParentObjectDelete on CustomObject__c (before delete){
List<Id> idsToQuery = new List<Id>{};
for(CustomObject__c a: Trigger.old){
    idsToQuery.add(a.id);
 }

//query all child records where parent ids were deleted
ChildObject__c[] objsToDelete = [select id from ChildObject__c where ParentId__c IN :idsToQuery];

delete objsToDelete; //perform delete statement
}

This is if you are planning to do a real time delete .Also in this case be sure to check with business what if someone undeletes a record from the bin.May be a trigger to handle this as well

Also if you don't need a real time delete ,a batch process to fetch orphaned childs(ParentId is null) and delete .