APPEAL: Piers' parents Roger and Dawn Hopson
Roger and Dawn Hopson of Melfort Road in Crowborough admitted on Wednesday putting up the money was "sheer desperation" after their 35-year-old son disappeared on January 25.
Piers Hopson, whose Asperger's syndrome means he would not be able to approach anyone for help, left his St Leonards-on-Sea care home with just over £40 and a cash card on January 25.
His mother Dawn Hopson, 63, told the Courier: "He must be staying somewhere. Piers could not survive on his own, someone must be helping him in some way or sheltering him and we just want this person to get in touch and let us know how he is.
"We just want to say thank you very much for looking after him and we do appreciate it, we are sure it is with the best intentions.
"But we love him and miss him very much, and want to know where he is and that he is safe and well. We thought the money might make someone pick up the phone and say, 'Yes, he is staying with us at the moment'. He can't have any money by now because it is now over two weeks and that is all he had to survive. We do thank them, but please telephone police and let us know where our son is."
He left his care home in Southwater Road at about 3pm saying he would return later.
It is understood he got on a train at St Leonards Warrior Square at around 3.50pm and got off at Hastings station.
CCTV released by police this week shows him walking in to Devonshire Road five minutes later.
Heading the Sussex Police investigation is Detective Constable Paul Wren.
He said on Wednesday officers were focussing on key lines of enquiry, which did not include murder.
These were following up reported sightings and examining further CCTV in those areas, investigating travel networks, holding a further leaflet drop and creating a time-line of sightings.
Twenty-seven people had contacted officers with possible sightings and that number continued to grow, he said.
They were in Tunbridge Wells, Heathfield, Hastings, Eastbourne, Horley, Crowborough, Fairlight, Chichester, Canterbury and Covent Garden.
He said: "We would not discount any [sightings] because of Piers' propensity to travel because he has a bus card and a rail card, so that is another line of inquiry, to make further enquiries with the South Eastern regional transport links because it is feasible he could have travelled anywhere."
He said some information put forward appeared not to match Piers' character, including a possible sighting of him begging and of a man in a red anorak.
Asked if someone could be holding Piers against his will, Det Cons Wren said: "You can't really rule anything out."
"Purely at this moment in time we are focusing on the enquiries I have mentioned. We are going out today [Wednesday] with a questionnaire to jog anyone's memory, focused in the St Leonards-on-Sea area."
If you find Piers call 999 or Sussex Police on 0845 60 70 999, quoting serial 1721 January 25.