You don't have to get rid of street parking entirely. Another option is to put a separated bike lane between the parked cars and the sidewalk. This way cyclists are protected from traffic and parked cars (ensure a bit of room for the passenger door).

Also the increased cognitive load of having to pay attention to many things occurring on the road is not necessarily a bad thing. There is evidence that having to pay attention to many things forces drivers to slow down, which is safer and results in less injuries and deaths. In contrast wide lanes allow drivers to feel comfortable which increases their speed and results in more dangerous roads for pedestrians.