Our group was in that situation.
One leader, drove the change. Now there is a completely different ethos. It took him being 'difficult' and an 'arsehole' to a couple of parents. He was more than reasonable in terms of how much he was putting in, in dictating to other parents that he wasn't their doormat or babysitter, though not all of them saw it like that!
I remember speaking to one unhappy parent, who didn't know I was friends with this leader. Her son got the boot because she never ever did anything to support the group. She was blathering on about how unreasonable he was and how she didn't have enough time to volunteer. She worked part time as a mid day assistant at primary school within 3 minutes walking distance. He worked full time (in the next town over) and was doing 2 or 3 nights a week just running scouting nights never mind the admin. It just doesn't wash as a reason. She simply couldn't be arsed. The leader, simply recognised that and acted accordingly. Its tough on the kids but otherwise its unsustainable for the whole group.
Once he got one other hard core volunteer they were able to build from that, by being completely uncompromising about parental help. Parents were TOLD, not asked what they would be doing if there were no immediate volunteers. It they didn't work it out then they were put on warning and eventually booted. It wasn't negotiable and he made a point of saying he was just a parent and if he was stepping up to do all this then they could do at least something. Now the group has a reputation for being run by parents, its putting off families who aren't up for doing something to support the group which makes it easier as parents have a much better attitude from the word go. They DO make exceptions for particularly vulnerable kids - but to be able to do that, they HAVE to have all the other parents pull their weight.
This is why I stress its NOT a lack of time. Its a question of attitude in almost all cases and this has to be made crystal clear to parents in an uncompromising way. Leader had to be utterly hard nosed and thick skinned about it to get to where we are now. There wasn't an alternative. If he gave an inch, parents would take a mile.
It CAN be turned around. But its hard to do.