The Match of the Day host wants the newly appointed Crystal Palace to take a different approach to the FA Cup.
Pardew became notorious for selecting weakened sides in cup competitions during his time at Newcastle United, who have suffered miserable results in knockout competitions in recent years.
This term, Pardew reached the quarter-finals of the Capital One Cup with Newcastle but a 4-0 thrashing at the hands of Tottenham may have been a big factor in his decision to ultimately leave St James' Park for Crystal Palace.
And Lineker has urged the 53-year-old to do at Palace what he didn't at Newcastle.
Alan Pardew has got off to a good start. Be interesting to see if he's encouraged to give the Cup the respect it deserves at Palace.— Gary Lineker (@GaryLineker) January 4, 2015
Crystal Palace were winning at Dover at the time of Lineker's tweet - not quite Premier League opposition - but the sentiment still applies.
Newcastle never really got near cup success during Pardew's time at the club; Eagles fans will no doubt hope for something different.
That said, with Crystal Palace currently battling against relegation, there will be a lot more sympathy for the former Newcastle boss in south London should he decide not to prioritise knockout competitions.