Larvae eat larvae, adults eat adults. Yes, it’s a dog-eat-dog world out there.
Dragonflies and mosquitoes both live as adults for only a couple of weeks. But dragonflies spend several months or even years in the larval phase and mosquitoes only several days.
If you kill off the adults of both species, which will bounce back faster?
Mosquitoes, obviously, because their eggs, larvae, and pupae will all become adults in a few days, while those phases of dragonflies are still stuck in the water.
So an indiscriminate pesticide helps mosquitoes by knocking out a slower-breeding predator!
For a 50-minute video on the life and history of the drangonfly, see YouTube. Still from that video showing dragonfly larva about to munch into a mosquito larva: