whenener you update the contact if contact status is complete and its Account status is also complete then update the contact flag field to true

try with below code.

trigger contactUpdate on Contact(before insert,before update){
set<id> accids = new set<id>();
for(contact con:trigger.new){
if(con.status__c=='Completed' && con.AccountId!=Null){
accids.add(con.AccountId);
}
}
map<id,account> mapaccidwithvaules = new map<id,account>();
for(account acc : [select id,status__c from account where status__c='Completed' AND Id IN:accids]){
mapaccidwithvaules.put(acc.id,acc);
}
for(contact con:trigger.new){
if(mapaccidwithvaules.containskey(con.AccountId)){
con.flag__c = true;
}
}
}