The reason for wanting a cycle lane in the first place is to get out of the way of traffic which has a lot more mass/speed/momentum than a squishy person on a bike.
I reckon you can tackle this problem in several ways.
Firstly residential streets with car parking can stop being through routes to somewhere else. Just make it possible for local residents to drive or cycle to/from their houses. Discourage anyone else from using that street. For instance by blocking off one end of it with bollards (which people can cycle through).
Secondly, slow down the remaining traffic to a reasonable cycling speed, say 10mph. Do that not just with "10mph" signs but by redesigning the road so that it looks like a 10mph road rather than a race track. The Haymarket Colonies has done this quite nicely by putting giant plant pots in the road.
As soon as there's only a little bit of local traffic, and it's going at the same speed as a bicycle, there shouldn't be a safety problem.
If the road has to be a through route to elsewhere, then vehicles with vastly different momentums will be sharing it, so there's a basic safety problem. That's when it needs proper segregated cycle lanes - by which I mean that it should be pretty much physically impossible for motor vehicles to get into them or block them or affect them in any way. If there's room for car parking as well, go for it. It won't affect a proper segregated cycle lane in any way.