According to both the Oxford & Collins English Dictionaries a parish has several meanings.

Within the Christian Church a parish is a small administrative district or sub division of a diocese typically having its own church and a priest or pastor.

A civil parish is the smallest unit of local government constituted only in a rural area.

In some parts of the USA & Canada a parish is a territorial division corresponding to a county in other states.

A parish is also the area surrounding the parish church and the people therein.  This includes churches of other denominations other than Church of England, businesses and residents.

Related items

Parish Chest is what it describes basically a secure chest or cupboard where a priest or parish clerk would store important letters and documents sometimes even the parish registers and the church plate.  Very often too the log books of church business such as  poor relief accounts and even bastardy and removal examination documents can be found amongst the items stored away for safe keeping.  Sometimes too copies of parishioners wills were hidden away here especially if the priest was named as an executor or if the church were to benefit in any way.

Parish Clerk is a person designated to assist in various church duties, however in the case of an on-line parish clerk it is an advisor for the area within the parish boundary.

Parish Council is the administrative body of a parish whether it is parochial or civil.

Parish Log Book these were often kept but not mandatory.  Basically it is a diary of parish events and not many can be traced or are useful.  One that I personally have seen for a parish in Monmouthshire was very detailed with home visits for private baptisms that were not then transferred to the baptism registers.  It also contained notes about births in the village and some were even speculative as to the paternity of a child.  A true find if you come across one.

Parish Pump is an item of only local interest whether civil or parochial.

Parish Register is book kept within the church or with the priest that records baptisms, marriages and burials within the parish church.

Parish Relief or On The Parish is a welfare handed out to the poor of the parish to enable them to live from day to day.  This was adminstered by the church under the poor law rules.  Only people born within the parish could apply for assistance and for the poor of other parishes removal orders were often made and the person sent back to the parish of their birth.  In some cases the parish would help a non parish born resident but only if the home parish were willing to cover this cost.

Parishioner is someone living within the parish boundary or more correctly for poor law jurisdiction someone who was born within the parish boundary.

Parochial means that is it related to the church of the parish.