I cannot imagine why a parish would NOT provide for a basic right such as anonymous confession.
From the website of Catholics United for the Faith:
Regarding the option to confess behind a fixed screen or face-to-face, the Code of Canon Law states: “The conference of bishops is to issue norms concerning the confessional, seeing to it that confessionals with a fixed grille between penitent and confessor are always located in an open area so that the faithful who wish to make use of them may do so freely” (Canon 964 § 2). The Pontifical Council for the Interpretation of Legislative Texts provided more clarity to the norm in its 16 June 1998 decree. In this decree, the Holy See preserves the right of the priest to require confession behind a screen.
The relevant canon is 964 § 2 of the
current code of canon law.