Such as the question states it is possible to better method to write this:

SELECT clients_lists.*, 
       COUNT(clients_lists_relationships.clientid) 
FROM   clients_lists 
       LEFT JOIN clients_lists_relationships 
         ON clients_lists.listid = clients_lists_relationships.listid 
WHERE  clients_lists.parentid = 1 
GROUP  BY clients_lists.listid; 

Not necessarily. The query looks fine in my experience. You can test to make use of aliases for better readability:

SELECT c.*, 
       COUNT(r.clientid) 
FROM   clients_lists c
       LEFT JOIN clients_lists_relationships r
         ON c.listid = r.listid 
WHERE  c.parentid = 1 
GROUP  BY c.listid;

However it will not really create a technical difference (not too I understand).

Also make certain that both clients_lists.listid and clients_lists_relationships.listid are of the identical datatype and length and therefore are indexed. This can certainly help your query's performance.