Mosquitoes do get affected by the parasite; I'm not sure for Dengue, however, with Plasmodium they confer fitness disadvantage to the mosquitoes (many published reports are available). Nonetheless, if virus has to get multiply in the mosquito then it need building blocks and other resources which they draw from mosquito itself (Parasitic relationship). Moreover, either dengue virus or Plasmodium does not have a mutualistic or symbiotic relationship with the mosquito vector.
Secondly, these parasites have evolved to invade the bigger size organisms for their free multiplication. Hence they might have adopted to infect the larger animals which could provide them huge reserves of resources (comparatively to the mosquito). Biology of mosquito and vertebrates are different; hence they choose to invade the vertebrates, larger animals. These parasites take route of mosquito as a carrier which supports their multiplication and further infections.
Third, if they harm mosquitoes then mosquitoes may not live long enough to infect further. So they restrict their damage (evolutionary adaption) to the vector as compared to the ultimate goal, human/bird/animals, etc
Additionally,
Malaria is caused by single-celled Plasmodium parasites. When a person is bitten by an infected mosquito, the microbe makes its way through the host’s bloodstream to the liver where it replicates, returns to the blood and infects red blood cells.
But mosquitoes don't have red blood cells – their blood is a somewhat colourless fluid containing the blood cell equivalent called haemocytes – so the parasites instead escape to the mosquito's saliva from the gut.