You get 6 weeks of no job searching before EDD and then 6 weeks after EDD as well. So 12 weeks total of obligation free payments.
Get your partner off parenting payment and get yourself on there, and make him go on Newstart and get him to claim Exemption. Tell him to harass them for exemption, they'll end up sending him to Psychologist or some other thing and that buys you several months of no appointments etc. Centrelink is used to people trying to fraud the system, but as your partner legitimately can't work, just get him to push them to listen. There is a way for both of you to get payments, they just pretend there isn't for ages until they realise you're not lying.