it's definitely only possible for one parent to get "primary carer" status for a child/children. if you have 50/50 shared care, it would then look at your financial situation to see who has the greater need for the additional support. there is no way to share the PCC, or the single parent payment. if your incomes are low enough to qualify for parenting payment chances are that your income is low enough for your income estimate to give you the ftb health care card. yes, it may seem unfair, but if income support payments were split, at what point does it become workable/unworkable? if it is allowed for 50/50 care, should it be allowed for 60/40, 70/30?
income support payments are a form of welfare to help those who can't support themselves. given you have 50/50 care, there is a significant amount of time your children aren't in your care, so the theory is that you have ample time to be able to work and support yourself and your children, with the aid of the family tax benefit. if you're on a very low income, you can look at options like newstart to supplement your income...