Comparisons were drawn between Roy Hodgson's men and the Champions League winners even before they set out for Poland and Ukraine, with both having been written off after being forced to parachute in new managers.
But the fairytale turnaround masterminded by Chelsea boss Roberto Di Matteo is seemingly being mirrored by Hodgson at Euro 2012 after England defied the odds to top their group and set up Sunday's quarter-final against Italy.
Tottenham midfielder Parker said: "I think we have a real togetherness and commitment in the squad and, ultimately, that can get you a long way.
"We've seen that last season with Chelsea and other teams.
"Fundamentally, they are the main key ingredients of why we can go and progress further.
"It's going to be a very tough game against an Italian side that are very well prepared and organised, with individual qualities.
"If we carry on doing what we are doing, we can maybe upset them."
Di Matteo made an art form out of causing upsets in one of the most astonishing European Cup triumphs ever.
That was largely down to the incredible team spirit he fostered, something which had been lost under sacked manager Andre Villas-Boas.
England's players have been queueing up to hail the togetherness Hodgson had engendered in their camp, with Parker admitting the more relaxed approach had been well received.
As when Chelsea staged their remarkable fightback to beat Napoli in the Champions League last 16, England could hold their heads up high if reaching the knockout stage proved the limit at Euro 2012.
But, like the Blues, they are not going to be satisfied with simply saving face in Poland and Ukraine.
"We want to go further than the quarter-finals," Parker said.
"We have the belief in the squad to go and get a victory at the weekend and progress.
"We'd be really disappointed if we went out of the competition on Sunday."
That disappointment arguably would have been less had England finished runners-up in Group D and drawn Spain.
"I don't think I am being disrespectful by saying we'd rather play Italy than Spain," Parker said.
"Spain are the reigning world champions and European champions."
Sunday's game in Kiev could see Parker come up against Mario Balotelli for the first time since the Manchester City striker was banned for stamping on his head during a Barclays Premier League game in January.
"Whether Balotelli meant to do what he did or not, I don't know," Parker said.
"He got punished for it and obviously has served his punishment.
"I am not holding any grudges."
Parker is well aware of Balotelli's firebrand reputation but refused to be drawn on whether the Italian was in danger of wasting his talent.
But the Spurs star was under no illusions what effect that talent could have when channelled in the right direction.
"Of course, he's a threat," Parker said. "We have all seen his quality.
"I can only really comment on his footballing ability. He is one of many in their team that we will have to take a tough line on."
Penalties have been a huge talking point in the build-up to England's first knockout game but Parker's name may not be the first on the list to take them.
"I have not taken many penalties in my career - I think I have only taken one," he said.
"I did score, yeah. I think it was in the cup for Charlton.
"It's not something I am familiar with.
"At the same time, with the preparation and planning that goes into it before the weekend, I'm sure if needs be, I'll step up."
England were expected to practise penalties during today's training session, which saw all 23 players take part.
That included Ashley Young, who suffered a knock to the shin in Tuesday's Group D win over Ukraine, and Jermain Defoe, who is back with the squad after flying home to attend his father's funeral.
It was impossible to glean England's likely starting line-up from a practice exercise between those wearing bibs and those not during the open part of the session, with players who started against Ukraine scattered among the two teams.