This must be so frustrating for you. I agree that in any voluntary enterprise, there are always a few who step forward, over and over again. @BrownTurkey has a good, point, in that instead of looking for individuals to help, it would be great to get a small core to organise the rota on your behalf, well on behalf of the children (and their parents) who want to take part.
They could listen to any concerns about what would be required and organise the rota so that everyone has to volunteer at some point, adjudicate any exceptions (younger DC, SN etc) and ensure that there someone on standby, or more than one volunteer each evening. You could get agreement from the small core on how it will run and how people will be told what they are required to do. So often, a raw volunteer will not really be much use and will find the experience unsatisfying. Typically, the more you do the more you can put in and so get more out of it. You could be very clear, someone required for first x minutes to do x (take the register, help with coats, set out equipment, etc) Someone around for the whole period, they will be do x, y and z. Someone required at the end to oversee a clean up and lock up. It might be useful to have some flexibility so that some parents might be able to stay for the beginning for a whole term, others prefer to do one whole evening a term.
Communication between organisers and helpers is so important to smooth running and the feeling that everyone is getting something out of it. Perhaps start setting it up over the summer. There could be a launch of the new service, when you would explain what is required and at what time.
People are more likely to volunteer or keep volunteering in the places were i can see i am contributing, rather than the places where overworked organisers are too busy to explain what they need me to do. Things they have been doing for years and don't realise they still need to explain to someone new. Your core group could take over the induction for new parents, new people could be parent with experienced helpers.
All the best, hope you get the support that is required