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?
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”, who also offer other possibilities:
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.
