India had opted to drop allrounder Hardik Pandya and play debutante Hanuma Vihari, who bowled only one over on the opening day.
Asked if India missed a fifth bowler, Bumrah said: "I don't know about team selection. That's a question for the management.
"When you have an extra bowler, that gives you some cushion while bowling. With four bowlers you tend to bowl more overs because then you have to come back quickly (to bowl).
"That was the only difference I felt, otherwise we tried our best, we bowled our hearts out, we bowled a lot of overs. An extra bowler sometimes gives you enough rest," he said.
England were placed at 198-7 after day one but they escaped to 332, thanks to Jos Buttler's 89.
At end of day two, India were placed at 174-6.
Bumrah said: "We were in a good position at 190-odd for seven, but they batted well and we couldn't capitalize. So it is a combination of both. We tried hard to bowl in the right areas but we couldn't bowl that well today and they also applied well."
England's lower order has been a problem for India throughout this series, and it was once again the case on Saturday as Buttler put on 33 runs with Adil Rashid and then 98 runs with Stuart Broad.
Asked about any specific plans for the lower-order batsmen, Bumrah said, "There has been no specific planning for lower order batsmen. You plan for each batsman, even if they are lower-order, we respect them. We tried to execute plans today but it didn't work.
Talking about Buttler, Bumrah said that the batsman took his chances and it paid off.
"Just because Buttler scored runs, doesn't mean we found it difficult to bowl against him. As a bowling unit, we never say if we get him early that we found it easy. He played well today and was taking his chances," he said.
"When you are batting with the tail, you can take your chances, take your freedom and our batsmen have done that as well before in the first Test when Virat (Kohli) was playing with the tail.
"So you have a lot of freedom and you can express yourself – you have nothing to lose. Nobody would say anything if you get out, so a batsman has win-win situation in that position. If it doesn't pay off, it's okay; if it does pay off it's a bonus for the team," he added.