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.

In a game against the Yankees less than a week after he openly criticized the team, 25-year-old pitcher Hunter Dobbins delivered a standout performance, pitching six scoreless innings to help the Red Sox secure a 4-3 victory at Fenway Park. Earlier, Dobbins had boldly stated to the Boston Herald that he would prefer retirement over signing with the Yankees.

This week turned out to be quite eventful for Dobbins, a promising rookie who instantly became a focal point in the heated Red Sox-Yankees rivalry. His remarks about disliking the Yankees added fuel to the fire, especially as he recounted how his father claimed to have been drafted by the Yankees twice and associated with Andy Pettitte.

However, The Post’s Joel Sherman discovered the truth: Dobbins’ father never played for the Yankees or any other minor league team, and he had no connection with 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.