Spread of malaria from an infected person
With humans infected with malaria, the parasite load in the blood can be quite high. It ranges from 10 to 108 Plasmodium falciparum parasites per ml . 

When a female Anopheles mosquito bites an infected human, they take in infected blood, not just the parasite. They take in a few micro liters of blood, which in usually enough to get themselves infected, and the next time they bite a human (about one week later), the infection is spread. 

You can learn more at the CDC malaria FAQ and their overview of the malaria transmission cycle.