You need to distinguish the function of a parish as part of the Church of Scotland, and its part in civil administration and local government.
Yes, Tarbolton is a historic parish. See
https://stataccscot.edina.ac.uk/static/statacc/dist/parish/Ayrshire/TorboltonAll the old Church of Scotland records are on the basis of the parishes, which is why you would find information about people in these villages in the Tarbolton parish records.
When censuses began in 1801 the collection of the information was also organised on the basis of the traditional parishes.
In 1855, when statutory civil registration began, the registration districts in rural areas were identical to the parishes. (The cities were subdivided into a variety of RDs. Some, for example in Aberdeen, corresponded with the historic parishes, while some, for example in Glasgow, did not.)
Over time changes were made to the boundaries of the RDs as populations changed, and some RDs were subdivided.
Parishes were also used for compiling the Valuation Rolls and the Registers of Electors, though again the boundaries were adapted to population changes.
The Poor Law was also administered by parish from 1845 until it was superseded in the 20th century.
There were also changes to the parishes themselves. Even as early as the 17th century small parishes were amalgamated to make bigger ones, and in about 1891 there was a great tidying-up exercise that did away with some of the more illogical boundaries.
In 1975 all the county, town, burgh and city councils were abolished and replaced by new local government areas. However the Valuation Rolls were still collected by parish after 1975 until they were in turn abolished, and the Registers of Electors are still based on the boundaries of the parishes which, it is wrongly claimed, have been abolished.
(The Ordnance Survey does not help matters because it has deleted the parish boundaries from its present range of maps.)
There was also a shake-up of RDs at about the same time, followed inevitably by the closure of little-used registrars' offices in small communities.
So yes, Tarbolton is a parish but no, apart from providing the geographical basis for the electoral register it no longer has any function in public administration. It may still be used by the Church of Scotland as its unit of organistion - you'd have to ask the minister about that.
