Not getting into the debate, but just posting on the math fail.
If there are 1000 people consisting of 500 males and 500 females, there would NOT be 1000 "pairings." Assuming one male + one female makes a single pairing (it takes 2 people to make a pairing), there would be 500 pairings consisting of 2 people each. So if each pairing had 2 children, that would be 1000 children.
1000 = 1000.
If every couple in the world had exactly 2 children who both lived long enough to have children of their own, the world population would be the SAME, not increased. The issue is that a lot of families have way more than two children, particularly in poorer parts of the world where more children = more help on the farm, or where most children previously wouldn't live to adulthood but modern medicine means they would.
As for reincarnation, not all reincarnations are said to occur on the human level, or even among animals. Really good people are reincarnated as sorta demigodly figures above the physical plane we live on, and really bad people are reincarnated in some kind of a hell. So an increasing population could mean that more souls in hell are becoming good enough to join the human plane and/or more demigodly souls are being bad enough to be reincarnated as humans.
It's not my personal belief, but if someone's going "reincarnation is impossible due to the increased population," then according to the beliefs of religions that include reincarnation, no, the increased population does not make it impossible.