The Rockets took a 43rd minute lead at the New Peter Mokaba Stadium on Saturday through a Jean Lwamba Lubumba own goal.
Mfanafuthi Mkhize equalised in the 58th minute and Lubumba made up for his earlier error with a 71st minute winner.
Parker had only been in charge of the team for a week in the lead up to this game, following the sacking of Adnan Beganovic.
It was always going to be a tough ask for the inexperienced mentor to lead Galaxy to a second Nedbank Cup title following their 2019 triumph over Kaizer Chiefs.
Speaking after the encounter, he said he felt the team performed well but were undone in the second half.
“It’s a tough one; it’s a tough one. First half, we had a plan, we stuck to the plan in terms of making sure we neutralise their threats and their runners to run behind our defence. We did well in that regard, us coming out after regaining possession going forward, we could’ve done better in that regard,” Parker told SuperSport.
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“But the players didn’t have the calm aggression in terms of our forward play and being in possession. There and there were moments, but we needed more of the moments in that regard and set-plays we’ve been working a lot on set-plays it’s where we got our goal from a set-play.
“We posed a threat in that way which is good, then we went into halftime, we had a strong chat halftime that we cannot turn away with this opportunity that God has given us and we must make sure we got in and show more character at the same time enjoy the match.”
Parker added: “We could’ve got something but it’s football, and we just have to make sure we work hard for the rest of the games and collect as much points as we can. Also, congratulations to Durban City.”