Determine a set of id's that exist in a set of tables: options needed to speed up or parallelize the algorithm
This belongs on dba.stackexchange.com  but I don't have the rep to comment
(and the reason I don't have the points is I gave out a 100 point bounty)

You don't give the sql an algorithm.  You give it indexes and a good query (some times with query hints).  SQL will parallelize on its own. 

PartyID would be better as an int and then have another table for the text
Int or VarChar put and index on it  

select table1.partyID 
  from table1 
  join table2 
    on table2.partyID = table1.partyID 
  join table3 
    on table3.partyID = table1.partyID


for an or just do do a union  

select table1.partyID 
  from table1
union
select table2.partyID 
  from table2
...