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. 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 and 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. You never know, this would be a dream, a fairytale after everything we have been through, to get there."