Going 3d with dedicated bike skyways can solve this problem, but bikes are 3rd/4th class citizens in most places because they get a relatively small fraction of overall transportation. (Cars, walking, subways or other public transit, then biking.)
Crossing a bike lane isn't the same as stopping in it. You can check that there aren't any bikes in the bike lane before crossing it to park. You can't check that there won't be any bikes in the lane for the next 30 seconds before stopping to load some groceries.
On top of that almost every bike lane I have seen has had parking one side with the only way to park a car is for the car to also use the bike lane. EX: 3rd google image for bike lanes: https://bikesiliconvalley.org/wp-content/uploads/Veterans-Bl...
Going 3d with dedicated bike skyways can solve this problem, but bikes are 3rd/4th class citizens in most places because they get a relatively small fraction of overall transportation. (Cars, walking, subways or other public transit, then biking.)