Parker, 38, takes charge of his first game as Fulham's caretaker manager tomorrow at home to Chelsea, but he has been planning his move into management since he started taking his coaching badges while a player at Tottenham Hotspur seven years ago.
And he feels Kepa's public show of defiance might have had an overall positive impact on the dressing room.
"For our confidence, it was important to play at the same level against City after the match in the Premier League".
Parker was relegated from the Premier League as a West Ham player in 2011 and admits the experience hurts.
Sarri appeared to be considering storming out of Wembley as Kepa refused to come off, but he returned to the sidelines after regaining his cool. The Italian said: "To take this job, you have to love football, but you have to forget everything that is around football".
"Yesterday - I don't really have many friends - the phone was just pinging away", said Parker. Now I can understand very well that, in England, it's very hard to be in the top four.
"He played with his team-mates well, I think that he needs more acceleration because now he is a little more resistant but without the acceleration he had in Naples for example".
"I'm sure at the end of the season, when all the dust is settled, everyone will sit around a table and say "OK, look, this, this and this wasn't good enough, this was perfect".
"I realise the task ahead is very hard, but it's not mathematically impossible so certainly it's something we'll strive to do".
Sarri believes Chelsea are finding some consistency in their performances now, which he hopes will continue.
"[Former Manchester United manager] Sir Alex [Ferguson] found the time to send a long, detailed message giving me advice on how he sees things, which is truly inspirational".
Alonso repaid his head coach's faith with his most solid performance in several months, and Sarri said he expects even his best players to go through hard moments. "But, in 11 months, you have to accept one or two months of difficulty".