What is the physical layout of the entry door that parents are using? Is it into a hallway or foyer where no one is necessarily around? It sounds as if that is the case. Is there an office for the preschool? So visitors do not enter through that office? It may just be physically impossible for that to happen -- at my child's preschool years ago (in a church) the preschool entrance was into a hallway and the director's office was down that hallway some distance. You can't necessarily change the layout of the building, but you should indeed talk to the director (not the teachers, this is a director issue and possibly an issue for the parents' board if there is one).
A doorbell would work well. The issue would be the preschool staff AND PARENTS sticking to a policy about who answers that doorbell, or else you'd end up again with parents just opening the door for other parents (or strangers) as the parents come and go.
Our schools here (not preschools but elementary) all have a doorbell and camera setup: There is a small box on the exterior wall by the front doors; for entry, you press a button on the box, which rings a doorbell on the secretary's desk -- and she can SEE who you are as you stand outside the door, thanks to a camera in the exterior doorbell box that has a monitor on the secretary's desk.
Yes, sounds expensive, and would depend on the preschool having a staff member who is always present at a desk to oversee the coming and going.
The preschool at the church we attend has locked doors at all times but the preschool office has large windows that are about two feet from those doors and the director and secretary can see all visitors from those windows and must "buzz in" people for entry.
All that said....It is fairly typical, I think, for many preschools to be casual, even lax, about comings and goings. Many are in churches or community centers or other buildings with other purposes, and are arranged architecturally so there is no office near the preschool entry door. This isn't good, I agree, but if you bring it up to the school, see how they react. Then weigh your concern against the many positives of this school. It does sound great. The key thing too is -- can KIDS wander out?