Parkin said: "Everyone was in massive shock when Alan was sacked. It has been quite strange.
"Alan brought me to the club and I have a lot to thank him for. We as players all feel responsible for him losing his job."
Wayne Rooney has returned to Everton after 13 years at Manchester United.
Germany defender Antonio Rudiger has joined Chelsea from Roma on a five-year contract.
Wayne Rooney has returned to Everton after a record-breaking career at Manchester United.