The Clarets had been forced to wait to get their hands on the silverware despite winning the division at Charlton on Saturday following the Football League's controversial decision not to allow teams to lift the cup at opposition's grounds.
Having used an inflatable version at The Valley, Sean Dyche's squad were able to celebrate with their trophy in front of the steps to Burnley Town Hall on Monday evening, but both defender Tarkowski and former England cap Barton, who were the last two introduced to the crowd, missed out on receiving their winners' medals.
Writing on their Twitter account, Burnley explained: "Burnley Football Club would like to apologise for the mix-up during the medal ceremony. (1/3) #Champions
"The Football League advised the club that 25 medals were to be presented, but the club put forward 27 players. (2/3) #Champions
"We would like to apologise to the @football-league for this administrative error and every player will be receive one. (3/3) #Champions"
A representative from the Football League had earlier been booed when introduced to the crowd before the players came out while Barton appeared uninterested in lifting the trophy when it was passed around following his snub.
Barton and Tarkowski both took to Twitter to play down the incident afterwards.
The latter wrote: "It was only banter, as if I'd refuse a Championship winning medal! The ribbon wouldn't fit over my ears so waiting for one with more give!"
Defender Ben Mee, who was voted players' player of the year on Sunday evening, expressed his disappointment at the Football League's decision to make Burnley wait to get their trophy.
"It was a shame we couldn't have it on the pitch on Saturday - everybody's had their say on it, it was ridiculous," he said.
"But we saw it last night and we got to hold it, so to officially have it is great and to do it in front of the fans was great.
"That's what we were upset about, the fans travelled to Charlton for an early kick-off, it was great support and a real shame we couldn't show them the trophy. I can't understand why because I don't think anyone was going to disrupt that."
Dyche's squad attended a civic reception in the town on Monday before former Labour spin doctor and Burnley fan Alastair Campbell interviewed several players on the balcony and they then embarked on an open-top bus to Turf Moor.