Parkinson’s side took the lead against Chelsea twice before ultimately succumbing to the visiting side’s quality.
Decisions didn't go Wrexham’s way, however, with George Dobson shown a straight red card for a rash challenge on Alejandro Garnacho and a late equaliser chalked off for offside.
Speaking to BBC Sport after the game, Parkinson was full of praise for his side for their valiant effort, saying: "I'm very pleased with the way we acquitted ourselves tonight.
“We showed a lot of good aspects to our play and played some great football. We weren't just coming here to stop Chelsea - we believe we can play.
"We were frustrated not to come in at half-time ahead. The lads have given a lot tonight and to play extra time with 10-men is really hard.
"We wanted to have a real go in the second period and we had some great chances. We had moments even with 10 men and the lads have to take a lot of credit tonight."
On Dobson's red card, he added: "In the second half I think we were the better team and I was confident we were going to win. The referee gave a yellow.
“I'm not sure of the rules, it's the first time we've had VAR and it has to be clear and obvious to overturn it. I think the referee could have just kept it at a yellow.
"I said at the break in extra time there are still going to be chances for us. Unfortunately we didn't take one but we have to take a lot of heart from that performance.
"It's been a great cup run. Tonight was a great spectacle and we were competitive against a very good side. We will take a lot of positives from it."