Yet Eric Dier and Son Heung-min struck in the second half to earn the points, leaving an encouraged Spurs manager Mauricio Pochettino to declare: "It was important for us to be there if Chelsea failed - and we are there".
"You start by setting standards and start setting out basic responsibilities on what you need to do to win a game of football".
Asked on BBC Radio 5 live's "Sportsweek" programme if Palace could find the "enormous pots of money" required, Parish replied: "Enormous?"
"Our performance was good but I don't want to use this word unlucky because I don't like this word". They have a really strong squad.
"For sure, for people watching this league, it's a good result". To go game by game.
"In England an easy game does not exist and the game is very tough if the name of your opposition team is City or another team".
Chelsea's position is still the envy of their rivals.
The Blues require 21 points from the final nine games to be certain of the title. You have to learn from these situations and try to win the next game.
"We have tried to keep this position".
Do YOU think Hazard will still be a Chelsea player next season?
However, despite being a goal to the good against Crystal Palace, the Blues fell to a shock defeat and have seen their points gap at the top cut to seven points. We all know the strength of their team. At £35,000-a-week, whether we think it's a lot of money or not, it is not in the Premier League.
The Frenchman has played in 26 of Palace's 29 Premier League games this season after joining for a then club-record fee of £10 million in 2015.
Conte told a news conference this week that he believes Hazard can satisfy all his ambitions at Stamford Bridge, with an improved contract rumoured to be on the table for the 26-year-old.
A second successive win at Stamford Bridge - after a 33-year wait for the first - saw Allardyce's men move to 31 points, further away from relegation danger.
Wilfried Zaha levelled the score line in the ninth minute from a low finish while Christian Benteke gave Palace the lead in the 11 minute after some poor defending from Chelsea before he dinked a shot over goalkeeper Thibaut Courtois.
But Palace are confident that Zaha, having resolved his global future with the Ivory Coast, will accept the challenge of leading the south Londoners up the table. I'm not concerned about this.
"Wilf is already realising his potential with the way he's playing".