The Premier League giants twice trailed to the Reds at a raucous Racecourse but battled back to reach the quarter-finals with a breathless 4-2 success after extra time.
With Hollywood owners Ryan Reynolds and Rob Mac watching, Wrexham took the lead through Sam Smith’s well-taken opener, but Chelsea levelled when a George Thomason clearance rebounded in off Arthur Okonkwo.
Callum Doyle restored the hosts’ lead before Josh Acheampong equalised again, sending the tie to extra time.
The visitors took control after George Dobson was sent off for a high challenge on Alejandro Garnacho, who then scored along with Joao Pedro to settle the tie.
Wrexham thought they might force penalties when Lewis Brunt found the net at 3-2, but the effort was ruled out for offside following a VAR review.
It was not the ending he wanted but Parkinson could not fault his players' efforts.
“The lads have been immense,” he said.
“Of course the dressing room is quiet, which you’d expect after a defeat, but when we wake up tomorrow the lads can be immensely proud – and the supporters will be – of the way we’ve played.
“Everybody knows that 11 v 10 in extra time, when so much energy has been used, is a difficult ask.
“But we said there would be a moment and there nearly was with the disallowed goal.
“I think we deserved, at the very least, to go to penalties, but some big calls went against us.”