Chiefs won 2-1 at Moses Mabhida Stadium in Durban to advance to the quarter-finals of the event, having also knocked the NFD side out of the tournament last year.
Parker had a hand in both Erick Mathoho and Castro's goals for the Glamour Boys, who needed a lift after a Soweto Derby loss to arch-rivals Orlando Pirates hurt their league title aspirations.
“We are delighted to progress to the next round,” Komphela is quoted as saying on his club's website.
“It wasn’t a walk in the park. Stellenbosch came out and really made a fight of it. However, before their goal, we had a couple of chances to bury the match, including Teenage Hadebe’s header. That was a clear-cut chance, but we failed to finish from close range.”
Komphela added: “There were a lot of positives to build on, even Castro’s goal came from Parker’s touch.
“We will take this win to the next match, because after a win one goes to the next match with less pressure.”
Meanwhile, Stellenbosch coach Steve Barker credited Chiefs' defence, singling out Daniel Cardoso for praise.
“Cardoso was fantastic tonight. No wonder that Chiefs have conceded so few goals this season, as they are very tight and well organised in defence.”
Bloemfontein Celtic, Maritzburg United, Baroka FC, Ubuntu Cape Town and Free State Stars are also through to the next round of the showpiece.
Mamelodi Sundowns tomorrow face minnows EC Bees and Cape Town City will host Orlando Pirates prior to Thursday's last-eight draw in Cape Town.
Komphela's side, in fifth place on the table, next entertain AmaZulu in the league at FNB Stadium this Saturday (18h00).