IMHO, you are not asking the right, initial question which is am I going to be able to get five award tickets on the same flight on the same airline to very popular destinations such as Paris?
Answer: probably not. You may have to split up the family and use two or more flights.
Also, more info is needed. When do you want to go. Summer much tougher than, say, Fall or early Spring.
There is no one "best" program to get miles in. As with many things it all depends. Although many would say there is a "worst" program--Delta.
Since you don't appear to have any AA miles might be a good idea to start applying with Citibank for AA cards--to increase your flexibility.
And, consider, the tough part of award tickets is usually to get over the Atlantic. Think about non-award options for internal Europe travel. Car rental, trains, discount airlines etc.