The promoted side, who sacked Claudio Ranieri and installed former England midfielder Parker last month, are 19th in the table with 17 points from 30 games and face a tough test against title-chasing Liverpool on Sunday afternoon (Monday morning AEDT).
Parker's tenure began with defeats by Chelsea and Leicester City but he is convinced the team can turn their fortunes around with eight games left, despite being 13 points off the safety zone.
"There's only so many more weeks I can go on saying I believe we can stay up, but one result can kickstart something and we have the quality to try and do that," Parker told a news conference.
"When you look at how many games we need to win I understand it'll be difficult but while there's still points it's possible. We have some very good footballers and we need to start getting the best out of them.
"The attitudes and standards are rising on a daily basis. They have a chance at the weekend to prove how good they are."
A win for Liverpool will help them leapfrog Manchester City into top spot and Parker conceded facing Jurgen Klopp's side was a daunting challenge.
"Liverpool is a team that is fighting at the right end, in terms of fighting for the championship," he added.
"They're deadly with the pace and the power they have which is frightening. They have a top class manager and have done very well. They're a team full of individual quality and are up there with the world's best."
