Nicola Payne's parents, John and Marilyn, have given a statement on the developments of the case.
Two men, aged 74 and 45 from Coventry, were arrested today on suspicion of conspiracy to prevent the lawful and decent burial of a body.
Her parents said:
"We have been kept fully informed by police of the actions at Purcell Road and we are aware of the time scales this may take.
"We have also been informed of the arrests today. As there is a lot of information for us to currently take in, we would please ask that you respect our privacy while this is happening.
"We hope the arrests and activity will allow anyone who may have important information to come forward and speak to the police. We can then finally find out what has happened to Nicola.
As well as the searches in parkland, police are also searching another address in Miles Meadow, Coventry in connection with the inquiry."