Rooney, 31, joined his boyhood club from United on Sunday (9 July) on a two-year deal and spoke of his desire to win silverware with Ronald Koeman's side. The Croxteth-born attacker won every trophy possible during his spell with Manchester, but his career at the Theatre of Dreams ended with him consigned to the substitutes' bench.
Parker, who spent five years at Manchester United, thinks Rooney is disappointed by his performances over recent seasons and will be looking to remind everyone of his vast quality at Everton, the club he made his name with as a precocious 16-year-old.
"I hope more than anything that he [Rooney] goes there and does himself justice," Parker told talkSPORT. "He is only 31 and I think he knows in himself that he has let himself down. He shouldn't be in the kind of shape or the situation he is at 31 given where he has been.
"Now he has gone back to where he loves, expectations are going to be high. He is going to be expected to go there and deliver and help the young players.
"Everton fans will be looking for him to do it and, if he doesn't get anywhere near to that, I think they will let him know it because, as one of their own, they will want him to go out there and make them feel proud."
Rooney swapped places with Romelu Lukaku, who left Everton for Manchester United in a deal that could eventually be worth £100m. But England's all-time goalscorer will not be the sole replacement for Manchester United's latest superstar.
Everton director of football Steve Walsh recently flew to Italy to talk to a number of clubs about potential transfers, and a report from Bosnian outlet Radio Sarajevo suggests that the Blues have heightened their interest in former Manchester City forward Dzeko.
The 31-year-old transformed himself from villain to hero in the Italian capital last season. After a lean first season, the Bosnian scored a whopping 39 goals in all competitions for Roma as they secured Champions League qualification and pushed Juventus hard in the race for the Serie A title.