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Unexpected token '(' error batch apex
global Database.QueryLocator start(Database.BatchableContext BC)
{
    String query = 'select id,ParticipantName__c,EpisodePoPEndDate__c from BPCIAe__c where id in :CHildParentMap.keyset() OR id in :ParentWithdrawaldateMap.keyset()  and (EpisodePoPEndDate__c=null OR PoPEndDate__c > TODAY)';
    return Database.getQueryLocator(query);
}

Unexpected token ( is thrown at the  Database.Querylocator.

You can't use any sort of dot reference in a dynamic bind variable. So anywhere you are calling .keySet(), you would need to cache that Set to be referenced directly.
Invalid
String query = '...WHERE Id IN CHildParentMap.keyset() OR Id IN :ParentWithdrawaldateMap.keyset()'

Valid
Set<Id> ids = new Set<Id>();
ids.addAll(childParentMap.keySet());
ids.addAll(parentWithdrawalDateMap.keySet());
String query = '...WHERE Id IN :ids'

A side benefit in this case is it allows you to eliminate an OR clause, which is always a win.

For your reference:

Dynamic SOQL Considerations
You can use simple bind variables in dynamic SOQL query strings. The following is allowed:
String myTestString = 'TestName';
List<sObject> sobjList = Database.query('SELECT Id FROM MyCustomObject__c WHERE Name = :myTestString');

However, unlike inline SOQL, dynamic SOQL can’t use bind variable fields in the query string. The following example isn’t supported and results in a Variable does not exist error:
MyCustomObject__c myVariable = new MyCustomObject__c(field1__c ='TestField');
List<sObject> sobjList = Database.query('SELECT Id FROM MyCustomObject__c WHERE field1__c = :myVariable.field1__c');

You can instead resolve the variable field into a string and use the string in your dynamic SOQL query:
String resolvedField1 = myVariable.field1__c;
List<sObject> sobjList = Database.query('SELECT Id FROM MyCustomObject__c WHERE field1__c = ' + resolvedField1);


