You can push a lot of water through a small and corroded water line, if your pump will delivr enough pressure. Changing the water line will not help with static pressure, which is pressure when there is no flow. Larger and smoother water lines will help reduce friction loss when there is water flowing. 20 PSI at static and 12 PSI with one sprinkler running tells me there is 8 PSI of friction loss in that pipe, even with just about 2 GPM flow.
I would replace the pump. Maybe go with a submersible that could be set at 80/100. This will give you 30 PSI more static pressure. Then if the pressure falls too low when running sprinklers or using water, you may still want to install new and larger lines. New lines first, will only give you 8 PSI more during use, and no extra pressure when static.