The Tigers nearly went a goal down seconds after kick-off, as slack defending allowed Andrew Johnson to break through and slide the ball to Jobi McAnuff, whose shot from 18 yards was tipped over by Bo Myhill.
It was not long before City did go a goal down - eight minutes, in fact. After Ben Watson's long-range free-kick floated dangerously into the box, Myhill did well to block Mark Hudson's powerful header, but he could do nothing about the rebound as Ward bundled it in from close range.
Myhill pulled off a superb last-ditch tackle to deny Johnson in the 22nd minute, but things were set to get worse for the Tigers. Two minutes later, Cort slid the ball into his own net from a dangerous cross by Wayne Andrews.
However, Palace's over-exuberant celebrations left them open at the back a minute later, allowing Parkin to reduce the deficit. He showed good control to chest the ball down and fire home from eight yards.
Jason Price nearly equalised five minutes before the break when he latched on to a Craig Fagan cross and fired in an excellent volley which flew just over the bar, with keeper Gabor Kiraly flailing.
But Palace always looked like they could increase their lead and City were lucky when the unmarked Johnson shot over seconds later.
Ten minutes into the second half, Cort nearly made amends for his earlier howler when his powerful ten-yard header from Stuart Green's free-kick went inches wide.
A few minutes later, a long-range effort from Parkin flew over the bar, as the Tigers began to wrestle back the greater share of possession. Manager Peter Taylor also threw on newly signed Darryl Duffy to bolster the strikeforce.
But the Tigers' sloppy passing was still giving Palace big in-roads. Johnson's 20-yard effort was tipped over in the 71st minute and the resulting corner caused chaos in the City box.
Lively winger Stuart Elliott made an instant impact on being introduced in the 74th minute, firing a snap shot at Kiraly, who managed to hold on to it.
Johnson was denied just a yard from the line in the 80th minute during another chaotic period in City's area, shortly before the referee booked Emmerson Boyce for wasting time before a throw-in, much to the delight of City's fans.
City tried their best during five frenetic minutes of added time, but it was too little, too late and still just not up to scratch.