But Parkinson says no further offers for his players have been made, and they would be rejected anyway.
He told the Hull Daily Mail: "I haven't heard of any bids coming in for Stuart so far.
"I haven't had any contact from Crystal Palace or any other clubs and if we were to, it would be a 'thanks, but no thanks' from our end."
Elliott, along with Barmby, signed a new contract recently. The pair both pledged their futures to 2008, and City seem determined for them to see that out. Barmby is rumoured to have turned down a move to Crystal Palace in favour of signing that new contract.
Meanwhile, Barmby, along with Danny Coles, Steve McPhee, Ian Ashbee and youth team member Ben Wilkinson, was away from the main session as pre-season training began yesterday. The five are still recovering from injuries sustained last season and will not travel with the rest of the squad tomorrow to the training camp in southern Spain. Ben Burgess and Sam Collins will go though, having returned to full training.