There was little Christmas cheer at the Hawthorns as Pardew saw a sixth Premier League game in charge end without victory, the stalemate only his third point since replacing Tony Pulis.
Both sides struggled to threaten throughout a drab 90 minutes and home striker Salomon Rondon wasted the best two chances.
Rondon was unable to test Jordan Pickford on either occasion, but his hamstring was stretched as he hobbled off early in the second half.
While Pardew waits for a win, Sam Allardyce's unbeaten start to life as Everton boss was extended to six league matches, the Toffees maintaining their place just inside the top half.
It is not so rosy for Albion, though, who are now 18 league games without a win and remain in the relegation zone, three points from safety.
The Baggies remain without a first-half goal under Pardew, but Rondon ought to have netted in the sixth minute, only to flick Jay Rodriguez's cross wide after shrugging off Michael Keane at the back post.
Albion continued to press and were again wasteful after 18 minutes, Craig Dawson - making his comeback after two months out injured - unable to turn Matt Phillips' dangerous cross in from close-range.
Everton also had a returning face in their XI, Yannick Bolaise involved after a year recovering from a serious knee injury, and the winger carried most of the visitors' threat, but chances on Ben Foster's goal were few and far between.
Instead, it was Jordan Pickford getting his gloves dirty before the break, the Everton keeper comfortably clinging on to Chris Brunt's speculative effort.
Rondon missed another glaring opportunity soon after the restart and it proved to be his final involvement, the striker limping off with an apparent hamstring injury after blazing wide on the end of Brunt's pass.
Half-chances were all either side could create as the tension mounted, Tom Davies blazing over for the visitors before Phillips was unable to convert a dangerous delivery from Brunt.
Albion piled on late pressure with a string of corners but the impressive Pickford comfortably saved from Hal Robson-Kanu before Phillips blazed a free-kick over the crossbar to sum up the Baggies' miserable run, before Foster was finally called into action to deny Oumar Niasse and ensure the points were shared.
KEY OPTA STATS
- West Brom extended their winless run to 18 Premier League games (D9 L9) – the longest such run by a Premier League team since rivals Aston Villa endured a 19-match run ending in January 2016.
- In contrast, the Toffees are unbeaten in their last seven Premier League games, winning four and drawing three.
- Everton have kept four clean sheets in their six Premier League games under Sam Allardyce - the same number they’d kept in their previous 23 in the competition.
- Former Everton man, Gareth Barry has played in more Premier League games on Boxing Day (18) than any other player in the competition’s history.
- Barry also picked up his 122nd yellow card in the Premier League, 22 more than any other player.
- Everton are unbeaten in their last six Boxing Day away games in the Premier League (W3 D3).