So then what is the difference between a mission and a parish? Do they not have mass at missions?
It might interest you to learn that a parish is a lot like a mini-diocese. In the early church the ecclesiology had basically one diocese-parish in a town, the pastor of all the town was the bishop. Other church buildings might be scattered around the town and in the hamlets beyond, where Mass is held, but for baptisms everyone had to go down to the cathedral where the bishop officiated. The diocese tended to be geographically small compared to modern north American diocese. This was the dominant organization in the Mediterranean area and basically remains the same in many places.
This is the reason for those old famous baptistries in Italy. Everyone went to the cathedral for baptism, the baptistry was a separate building outside, the local church buildings in the neighborhoods were not actually parishes. The diocese was not exactly separated into parishes although the church buildings in the neighborhoods would look like parish buildings to an American.
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As the faith spread into northern Europe the parish system was adopted. The land was sparsely populated and largely rural for a long time (some areas being essentially wilderness for hundreds of years in early Christianity), dioceses were geographically large, quite spread out, and functions like baptism were delegated to the local priest if far from the bishop. A parish could grow into several church buildings far apart in different local hamlets and villages (they might be called missions, but they are subsidiary chapels), with one parish priest in charge (under the bishop, of course, who would be even further away). The parish priest could do a rotation from village to village saying Masses, but if more priests become available they can be hired and assigned to the church buildings in the other villages. These are supervised by the parish priest who might be called the rector, these assistant priests are sometimes called curates or vicars or have a similar title in other local languages. Sometimes as the population rises over time a curacy is elevated to a parish, and more little missions and curacies might be set up in the smaller hamlets that have also grown.
Here is an interesting
short list of parishes in a town in Poland (where my family is from) some were always parishes, some are very old but served as local chapels for many years and were elevated to parishes later, so one can see how the process works.
If one by chance has ever seen the BBC television series Ballykissangel, set in Ireland, one can see this system in action. The priest in Ballykisssangel is a hired man, and the parish priest is actually in another village and is his boss.
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In the USA, the tendency has not been to have more than one church building in a parish, at least in urban areas. This means the one church building is often very large. This obscures the true nature of a parish system, which is like a diocese in miniature (because some originally episcopal functions have been delegated to the parish priest), and can cause some confusion to moderns.