Ferrari's Fernando Alonso made four stops on his way to victory in Sunday's Spanish Grand Prix, which featured 79 pit stops in total for the 22 drivers.
Pirelli said the changes were "made in the interests of the sport" and not to favour some teams over others.
Pirelli motorsport director Paul Hembery said: "We hope that (helping Red Bull) won't be the case, but we always face that risk.
"People will say it is pressure from Red Bull but there has not been excessive pressure from them."
On Sunday, Red Bull boss Dietrich Mateschitz was heavily critical of the current state of F1, saying it "had nothing to do with racing anymore". The Austrian met F1 boss Bernie Ecclestone, although he did not reveal what they had discussed.
Ecclestone said in Spain: "The tyres are wrong, not what we intended when we asked Pirelli to produce something which did half a race."
Hembery also denied that pressure from Ecclestone had informed the company's decision, saying: "He was only sharing the comments of the majority, that we had gone a step too far and we needed to come back a bit.
"So I wouldn't say it was pressure from him, it was really from the fans from a sporting point of view.
"From what we saw on Sunday, we felt, no, this was going in the wrong direction."
Hembery told Autosport they never intended there to be four-stop races.
F1 tyres, DRS and Kers explained
"We want to go back to having two or three-stop races," he added.
He said the problems had been caused by the performance increase of the leading cars, which are a second a lap faster than they were in 2012.
"They have basically been stressing everything too much, and probably we underestimated the performance," Hembery said.
Hembery said the changes to the construction of the tyres could affect the choice of the compounds Pirelli takes to Canada.
Last month, the company announced the two softest tyres would be used on the non-abrasive semi-street circuit in Montreal.
But Hembery said Pirelli would advise the teams this week of any revised plans.
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