It's claimed the pair met through their work for Canary Wharf College on the Isle of Dogs in East London.
The free school has previously been rated as 'Outstanding' by Ofsted.
According to the Daily Mail newspaper, parents are concerned the relationship between Stuart Cathrow and Laura Etherington could bring the school into disrepute.
43-year-old father-of-three, Mr Cathrow is said to have stepped down as chairman of the parent's association at the school. 46-year-old mother-of-two Mrs Etherington is reportedly no longer acting as treasurer.
On the Quaystone Church website, Stuart Cathrow is no longer listed as pastor and last posted a sermon online in May.
Parents of children at Canary Wharf College have reportedly claimed the pair are now living together and attending church together.
The principle, Sarah Counter, is quoted as saying:
"Neither the Reverend Stuart Cathrow or Laura Etherington are, or indeed have ever been, contracted employees of the school."
"This matter is of a personal nature between the adults involved and we respect their right to privacy in this matter."
Neither Mrs Etherington nor Mr Cathrow have yet commented on the allegations.