Warnock became the Premier League's first managerial casualty of the season when he was relieved of his duties less than 24 hours after Palace were beaten 3-1 by Southampton at Selhurst Park.
Friday's home reverse to Ronald Koeman's side, coupled with Hull City's 3-1 victory at Sunderland, plunged Palace into the relegation zone.
The south London outfit have won just three of their 18 top-flight fixtures this season, with their most recent victory coming against Liverpool some seven matches ago - and face a crucial trip to fellow strugglers QPR on Sunday.
Palace's poor form led to Parish and the club's board feeling they had no option but to part company with Warnock, who returned for a second spell in the hotseat in August.
Warnock's sacking means Keith Millen assumes the role of caretaker manager for the third time, after previously doing so following the departures of both Ian Holloway and Tony Pulis.
And Parish wants Palace's squad to respond in the correct manner and claim a positive result at the home of another team scrapping for top-flight survival.
"Hopefully the players will pick themselves up - there are no other alternatives," he told the Croydon Advertiser.
"We need a result and we need a reaction. Let's hope that's what they give us.
"All games are big and it's about three points. It's not bigger than that, it's important we get back on track and get some points. It's all we can do."
QPR sit two points above Palace in 16th place, with Harry Redknapp's team residing outside the bottom three solely due to their excellent form at Loftus Road.
Redknapp's men endured another miserable road trip on Boxing Day, as they were beaten 2-1 at Arsenal - their ninth straight away defeat in the top flight.
QPR are the only club in English football's top four divisions to have failed to pick up a single point on their travels this term, but it is a different story entirely at home.
The west Londoners have won four of their last five games on home turf, and will once again look to talisman Charlie Austin for inspiration this weekend.
Only Sergio Aguero and Diego Costa can better Austin's haul of 12 Premier League goals this season, and the in-form striker hit a hat-trick in QPR's 3-2 win over West Brom last time out at Loftus Road.
Redknapp may have to do without captain Joey Barton (hamstring), while Yun Suk-young (ankle) is a doubt.
Millen, meanwhile, looks likely to have something of a striker crisis on his hands, with the fitness of Marouane Chamakh (hamstring) and Fraizer Campbell (knock) unknown.
