The female mosquito (like female bedbugs) does the biting to provide food/energy to make the eggs she lays.

She mates only once, which is why releasing sterile males works to reduce mosquito populations.

https://www.rockefeller.edu/news/38495-what-we-got-wrong-abo...

the paper:

https://www.sciencedirect.com/science/article/pii/S096098222...