Deon Burton levelled from the spot, but veteran midfielder Graham Kavanagh notched a second-half double to secure a 3-1 win for the Cumbrians.
"We were beaten by a debatable penalty and two fantastic finishes," said Parkinson.
"I was annoyed and lost my cool, which is not good. The referee said there was a push but he almost had his mind made up before the ball came into the box. Any defeat is tough to take because we have huge expectations. We gave the ball away cheaply and got punished for it.
"Kavanagh scored two goals which were top Premier League finishes, they raised their game 15% to 20% and we didn't respond as well as we could have done."