Sources are never completely reliable or unreliable. It is the Historians job to evaluate the sources and from that reach drawn conclusions. However, different people, who may or may not see certain points as, for instance, biased, can interpret sources in different ways.
Both Parish records and census returns are public records, more so local records as they are kept in local record offices. They are made to keep account of the population and give other important information such as social structure, literacy and occupations.
Parish records, prior to 1837, would be kept by the parishes Vicar; they would record baptisms, marriages and burials. Basically the Vicar recorded all the ceremonies that took place in his church.
Other information that would be kept with the parish records would be; minutes of the vestry, registers of church services, records of church schools, tithe maps and awards.
The earliest parish registers were introduced to England and Wales in1538 and these would include baptisms, marriages and burials. This lasted until 1753, after this, marriages were recorded in a different register. Then in 1813 baptisms and burials also separated.
Everything changed in 1836 when the Government ruled that all births, deaths, and marriages should be told to local registrars who sent the information off to the General Register Office in London, becoming mandatory in 1874. It was called civil registration, and it meant the church now had no duty.
As in most sources there are factors in the history of parish records which make them unreliable in parts, on the other hand there is a lot of useful information which can be drawn from them.
Parish records were not always accurate especially during the Civil War, because of increased numbers of deaths and other factors that contributed to civil unrest. A Parish Register was appointed to record the necessary information during harder times such as the Commonwealth period, but some were nearly illiterate and/or weren't that concerned with their duties.