As for the government paid parental leave scheme - my understanding is that what is important is that you have worked 10 of the previous 13 months, at least one day a week. Whether you have a job to return to is not important (as it is meant to cover casuals and people in exactly your situation) ... this website has all the information on it ... http://www.humanservices.gov.au/cust...ntal-leave-pay. I am aware of several employers who have their own paid parental leave schemes where you have to return in order to get the money, and there are other centrelink benefits which have different eleigibility rules, but with the standard govt paid parentla leave scheme, having a role to return to shouldn't matter (although I think it can matter when it comes to who gives the govt money to you - i.e. whether you get it from your employer who is reimbursed by the government, or whether you get it direct from the government, but I'm not too clear on that point as I'm not usually required to give advice on the practicalities of payment, just eligibility for parental leave).
As to why the council won't keep you on their books just so you can get payments - its probably because they are concerned about that giving you rights you wouldn't otherwise have. For example if you are technically on unpaid leave that they approved, at any time you could say ""righto, I'm ready to come back now" and they'd have to either give you a job or terminate you and that creates risks for them. They would rather that you resigned and then they have no risk and don't owe you anything.
It's rough that you've found yourself in such a tough spot, I'm sure its the last thing you want to be dealing with during what is otherwise such a happy time - but hopefully it actually turns out to be a positive thing and you find something WAY better to go to instead!