OK, from experience, how to get people to help:
Firstly, make it friendly and easy to do. If it's a mysterious thing that happens before they show up and after they leave, it feels difficult and complicated. Make it visible. Ten minutes before the end, put the boxes out, ask each person to put away the toys that are near them. Our playgroup used to do a singing time at the same time to keep the toddlers busy, then parents who didn't need to be occupied with toddlers could help pack away.
Making it easy to do - is it obvious to people where they could park their babies while they help? Could you have an extra baby-watching volunteer? Three volunteers so one can watch kids and two can sort stuff?
Allocate things to people. Could this half of the room give us a hand packing up the drinks, and that half of the room give us a hand with the toys, if you can get away from your babies for a moment. They won't all be able to but some will.
Ask for specific finite help with the committee - we need someone to liaise with the room booking, it takes 30 minutes twice a year, could you do that? We need someone to count the money every week, you can do it in your own time at home, could you do that?
Ask for a fixed term. Ask someone to do something for 3 months or something. It's easy to think "I can't take on responsibility as I'm back to work in 6 months".
Bring the committee meeting to them. Make it part of a regular meeting not a need to get out in the evening.
Once people see in person how something works behind the scenes it feels much more approachable to volunteer.
Ask people in person. Get to know people. Pick on faces who are there regularly, say "I can see you've got your act together, how would you feel about ..." With any group there are inevitably people who feel at home, turn up every week, and stick around longer - often the ones who aren't going back to work. It's easy to think they don't mean me when there's a general call to arms. If someone asks you personally, not a hard sell, just asks - it can make people think again.
Just some ideas.