Instead, lefty Henry Owens, Joe Kelly and Wade Miley will start in New York.
Rodriguez will rejoin the rotation at some point next week, when the Red Sox return for a nine-game homestand.
Between Triple-A and the Majors, Rodriguez is at 140 2/3 innings this season.
"We're trying to manage his time and his workload, and this is a way to give him a couple extra days. It makes sense for us," said Red Sox interim manager Torey Lovullo.
Owens is at 146 1/3 innings this season.
"I think both of these guys are getting ready to pitch into their sixth month of baseball, and they've never done that before," said Lovullo.