Zaha was capped twice by England, in 2012 and 2013, but switched allegiances to Ivory Coast, the country of his birth, late last year and has since made six appearances for them.
He scored a fine solo goal in Ivory Coast's 2-0 friendly win over Russia last week, prompting Parish to tweet: "Never wavered in my belief of @wilfriedzaha, why is this goal not for @England?"
"I'm not sure it's helpful as a comment. I wouldn't comment on his decisions," Southgate told reporters after England's 2-0 World Cup qualifying win over Lithuania at Wembley on Sunday.
"I know the chairman's got quite a vociferous view on it. Steve's an intelligent guy who's done a great job for that football club."
Southgate, a former Palace captain, previously coached Zaha during his time as England Under-21s manager.
The Palace winger, 24, was the first player he watched after being confirmed as England's permanent manager last November, but by then it was too late.
"I didn't really appreciate there was this disappearing egg timer on him going to the Ivory Coast," Southgate admitted.
"I didn't want to just pick him because there was a chance that that might happen. The first opportunity I had as permanent manager, I did go and speak to him."
Despite his disappointment at missing out on Zaha, Southgate says he is only interested in players who have a burning desire to play for England.
"Any time you leave a player out of squads, you run the risk of them going across to another country," he said.
"But in the end I have to focus on the players that are desperate to play for England.
"If you don't feel that internal, 100 percent passion for playing for England, I'm not sure it's for me to sell that to you. It should be your desire to do it.
"The inherent desire to play for your country is the most important thing if we're going to be successful.
"Part of your identity as a national team has got to be pride in the shirt. So for me the commitment has to come from the player."