Question: When referring to members or worshippers at a Christian church, I might use the word 'parishioner', or possibly even 'congregation'.

Are there equivalent words specific to the Islamic faith for those who regularly attend or worship at a particular mosque?

Answer: 
A group of people offering prayers together is called a *Jamaat* or *Jammah*. Congregational prayer can be observed in any place as required, not necessarily a mosque; and a person joining a congregation for the first and last time is as much a part of it as a regular. 

Regarding *parishioner*: I am not aware of any formal term for regular worshipers at a mosque that would differentiate them from non-regulars. *Memberships* or *registrations* are not a feature of Islamic congregations.
The word *congregant* can be used, although it's not Islam-specific.

Actually this question came up on English.SE: [“Parishioner” vs. “congregant”](https://english.stackexchange.com/questions/87093/parishioner-vs-congregant), who also offer other possibilities:

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> To refer more generally to those attending services at a particular time, you could simply say *worshippers* or *attendees*; for all adherents, there are a variety of terms employed, such as *the brethren* or *the faithful*, or the more mundane *churchgoers* or *the observant*; *communicants* captures the sense of those in communion with the Church as opposed to outsiders.
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