The Glamour Boys face NFD outfit Stellenbosch FC in the last-16 of the event at Durban's Moses Mabhida Stadium on Saturday night.
With the recent Soweto Derby loss to Orlando Pirates putting a dent in their league title aspirations, pressure is intensifying on coach Steve Komphela to deliver silverware in the final few months of his three-year contract.
Parker is well aware of the stakes.
“We must do the business,” Parker told his club's website.
“The Nedbank Cup is an opportunity to win silverware for the club, the coach and ourselves as players.”
'Hond' expects a fired-up Stellies outfit, who are in mid-table under the watch of former AmaZulu mentor Steve Barker.
“They have a good coach and some fine players with a lot of energy. They usually play in a 3-5-2 system.
“We have to be in the right frame of mind. We need to be ready mentally, especially as this will be their biggest game of the season. Playing against Chiefs is for them like a cup final.”
The 31-year-old added: “We have to be careful not to put too much pressure on ourselves. We just have to keep it simple and take it one game at a time. Our focus must be going past Stellenbosch to reach the quarterfinal.”
In Chiefs team news, midfielder George Maluleka (ankle) and striker Leonardo Castro (foot) are doubtful for the contest with their respective knocks.