The reason, ostensibly, is the lack of park capacity generally available in Manhattan. Permits are pretty much the only way to reasonably schedule slots for all the schools serving the 1.6M+ people of Manhattan and their teams, since schools usually do not have sufficient play fields of their own.
During less busy times this isn't an issue. The permits more or less act as guaranteed reservations.
> If you are a member of an organized league, you must request a field or court for any formal use. Otherwise, our fields and courts are available to the public for informal use (pickup games). If you would like to reserve a field or court in advance at a specific time, you should make a request to ensure your uninterrupted use of the field.
During less busy times this isn't an issue. The permits more or less act as guaranteed reservations.
https://www.nycgovparks.org/permits/field-and-court/request/...
> When to Apply
> If you are a member of an organized league, you must request a field or court for any formal use. Otherwise, our fields and courts are available to the public for informal use (pickup games). If you would like to reserve a field or court in advance at a specific time, you should make a request to ensure your uninterrupted use of the field.