in the US, living with your parents after high school implies you need their support financially when you're old enough to have a job of your own.

Usually any mention of living with your parents implies being lazy ( the whole "neckbeard living in their parent's basement" stereotype ) unless you qualify it with "but I pay rent". If you pay rent then it appears more like you can support yourself and still not relying on mom/dad to pay your way even in adulthood.