I think there are some common themes that families should look for at any age, but otherwise this thread would be best served by being broken down by age level. The things that are important at 10-U and 12-U are different than at 16-U, and possibly different again at 18-U.
Generally, if I were a parent, I would look for certain things at any age level.
Professionalism - Starting with tryouts, how well-organized were the tryouts? Was there a tryout plan in place? Was the tryout done professionally? Was the communication thorough? Do the people running the organization and the particular team seem like they are on the ball? Are the plans for the fall and next summer specific, or is it just a general thought in the coach's head? Exactly when and where will the team practice? Is 100% attendance expected? If some players don't meet the practice expectations, what happens?
Roster - How many players will the team carry? (As far as I'm concerned, college exposure teams should carry 13-16. If you want to play college ball, why would you be afraid to compete for playing time? I guarantee you will have to do this in college, and if you aren't prepared mentally for it, you will likely quit before your junior year). If it's a college exposure team, who is handling the communications with college coaches, both at tournaments and in general? What tournaments is the team playing? If it includes invitationals, has the team been invited in the past? If not, what credentials will get the team invited this coming year?
Money - How much goes directly to team expenses and how much to the organization for organizational expenses? Do we get a rundown of team expenses so we can see how our money is being used? Who handles the money? Who decides how our money is spent, is it the head coach or the organization head? I could go on and on here, but I'd do some investigation and ask questions. Anything other than complete transparency can safely be considered a huge red flag.
Coaching - This is difficult if parents don't have a good deal of knowledge about the game. If I were a parent and wasn't sure whether a particular coach knew the game well enough to teach it, I would be asking around to as many people as I could find to get opinions. If I did know the game, I would simply ask the coach some philosophical questions about the game and maybe some game strategy and technique questions. You're up 3-1 in the 7th and the other team has runners on 1st and 3rd and no outs, what 1st-3rd play are you going to call if they steal? If they don't know the easy answer to that, then they shouldn't be coaching a competitive team. Do you teach hitting? If so, what method do you teach?
I could write a thousand more things here and any parent who asks all of this is likely to be seen as way too meddlesome, with their kid never finding a team. But these are examples of the things I would be trying to find out.