BOSTON — On the day before Father’s Day, Hunter Dobbins gave his father Lance the perfect gift: shutting down the team he never played for.

Facing the Yankees for the second time in a week, 25-year-old pitcher Hunter Dobbins delivered an impressive performance by shutting them out for six innings in a 4-3 Red Sox victory at Fenway Park. Just days before, Dobbins had stated to the Boston Herald that he would consider retiring if the Yankees were the only team offering him a contract.

This game marked a significant turn for Dobbins, transforming him from a promising young player to a central figure in the intense Red Sox-Yankees rivalry. Dobbins not only expressed his strong dislike for the Yankees but also mentioned that his father had allegedly been drafted by the team twice and was acquainted with Yankees pitcher Andy Pettitte.

However, a report by The Post’s Joel Sherman revealed that Dobbins’ claims did not hold up. It was found that Dobbins’ father had never been part of the Yankees organization or any other minor league team associated with them, and he did not have a personal relationship with Andy Pettitte.

On Saturday, Dobbins made sure his pitching took precedence over his father’s false credentials.

After giving up three runs — as well as a homer to Aaron Judge — over five innings in The Bronx in last Sunday’s win by Boston, Dobbins dominated the Yankees this time around.

He allowed just two singles and a walk in his 82-pitch outing, matching his previous best performance, when he threw six shutout innings against Kansas City on May 9.

Asked how it felt to beat Boston’s rivals again, Dobbins said, “It was a lot of fun, but I’m more worried about the win column — whether it’s against them or anybody. My job is to try to help this team win as many games as we can and play in meaningful games.”

The victory got the Red Sox back to .500 (36-36).

That meant more to Dobbins, he said, than the opponent.

“It’s more satisfying that we won the series and we’re building momentum and crawling back into this race,’’ the right-hander said. “There’s a lot of season left.”

Cora agreed.

“Our goal was to get back to .500,’’ the manager said. “It really doesn’t matter against who. There are a lot of ways to make it to October. Now we’re back to zero, neutral. We’ve got to keep playing well.”

Dobbins barely broke a sweat against the Yankees.

Trent Grisham reached on an error by second baseman Kristian Campbell to start the game before Dobbins struck out Judge and then got Ben Rice to ground into an inning-ending double play.

He pitched around a one-out single to Jazz Chisholm Jr. and a walk to Anthony Volpe in the second and whiffed Judge again as Dobbins cruised through the third.

The rookie retired eight straight after the walk to Volpe, before Volpe opened the top of the fifth with a rocket single off the Green Monster in left for just the Yankees’ second hit of the night.

Dobbins didn’t allow another base runner before leaving after six innings.

That included the top of the sixth, when Cora had Dobbins face the top of the Yankee lineup a third time and he delivered again.