The npower Championship side claimed an unexpected place in the semi-finals of the competition with a shock win over Manchester United at Old Trafford on Wednesday night.
The Eagles then managed to avoid Premier League big guns Liverpool and Manchester City in the draw for the last four and were handed a potentially easier tie against Championship rivals Cardiff.
Palace were beaten by Liverpool in the semi-finals of the competition in 1995 and 2001 but famously overcame Kenny Dalglish's Reds 4-3 to claim a place in the 1990 FA Cup final.
Parish feels another encounter against them would be great reward for the club after their battles with administration and relegation in recent years.
Parish told talkSPORT: "I am sure Cardiff were hoping for us.
"They'll feel they have got a chance and we were hoping for them.
"We could probably make more money out of Manchester City or Liverpool but we have been in one Wembley final in our history.
"It would be absolutely amazing for us to get there.
"No disrespect to Man City but I have got a soft spot for Liverpool.
"Kenny was manager when we beat them 4-3 in the semi-finals. You never know, this would be a dream, a fairytale after everything we have been through, to get there.
"But we have got a tough couple of league games in front of us and Cardiff will feel they can do the same."
A consortium headed by Parish and Martin Long brought Palace out of administration in the summer of 2010.
The club had also narrowly avoided relegation the previous season and were again close to the drop last term.
Parish feels they have now turned the corner under manager Dougie Freedman, who has made the team hard to beat and shown faith in youngsters such as the impressive Wilfried Zaha, Nathaniel Clyde and Sean Scannell.
Parish said: "We have invested in the club.
"The club has been starved of investment but we are not going to go crazy because we don't want the club situation ever again.
"We have made some buys but we want to rely on some of our youth.
"We want a blend of experience and youth to see what we can achieve and push on.
"Dougie is doing a great job. He is giving the youngsters a chance and he is coaching them well, schooling them and making them better."
Glenn Murray headed Palace's winner in the 98th minute at Old Trafford after a Federico Macheda penalty had cancelled out a wonder strike from Darren Ambrose to force extra time.
Ambrose's sensational 35-yard effort just after the hour ended a 550-minute goal drought for the London club.
Parish said: "It defied the laws of physics that shot, it was very good.
"Gary Neville was saying it was the best goal he has seen from an away side, which is a massive compliment."
Palace make a quick return to action as they host Derby in the Championship on Friday but the club are not complaining.
Parish said: "To be fair to the league and everybody, that was already organised as a TV game.
"We benefited from that from the TV revenue so we are not going to complain. We wouldn't have complained if we'd lost, we were party to that decision.
"Dougie and the guys knew what they were going to have to do.
"We are not going to use it as an excuse. Let's just hope we can carry the form into the league game on Friday."