Revd Jim Percival, 65, the vicar of Holy Trinity CofE Church in Freckleton in Lancashire, as well as his daughter, Ruth, 29, were first arrested in December 2014 after police were called to the vicarage following a report that a woman had given birth to a stillborn baby.
Police stated that they were following "numerous inquiries" into the death of the baby boy, and that it was being treated as suspicious.
Initially both were questioned on suspicion or murder as well as conspiracy to conceal the birth of a child, before they were bailed pending further inquiries.
However in January, Lancashire Police said a 65-year-old man and a 29-year-old woman from Freckleton would face no further action in relation to the allegations.
Outside the vicarage in Freckleton, Lancashire.
Since that statement the pair have been arrested and questioned on suspicion of child neglect.
Speaking on Tuesday, a police spokesperson said that the suspects had been further rebailed to April 13th 2016.
Revd Percival's wife, Susan, 66, was also held in December 2014 on suspicion of conspiracy to conceal the birth of a child, but three months later was told she faced no further action.
An inquest in Blackpool in April 2015 heard that the baby boy, named Jonathan Percival, was born in the early hours of November 25 2014 but was pronounced dead a short time later.
The medical cause of death remained unascertained as the full post-mortem report was awaited, the coroner was told.
A further hearing will take place at Blackpool Coroner's Court in March.
