Is It Permitted to Lock the Doors so That Parishioners Don’t Walk in Late?

There is nothing in Church documents that would address this. Given that some people may have legitimate reasons for running late it would seem to me to be counterproductive to lock them out. However that is merely my personal opinion. There could be safety issues that we are not aware of that the pastor is trying to address.

Given that there is nothing in church documents that addresses this the best course of action is to discuss the matter with the pastor of the parish. If his answer is unsatisfactory you could always raise the issue with the diocesan bishop.