The parent in question could apply for OPFP once separated 3 months as previously mentioned. If successful, the claimant would get OPFP plus half rate illness benefit. If s/he earns less than €146.50 then they would retain full OPFP and up to €425, s/he will receive a reduced OPFP payment. Means are assessed separately - as mentioned - first 20k disregarded. The advantage of OPFP over JSA is that you don't have to be available and seeking work for OPFP. On OPFP you may not get credited contributions which will affect future entitlements/pension.
But you can also receive the full OPFP and half JB (not JSA) if available and seeking work. Basic rates of payment for personal rate and child dependent rates for JB, OPFP and illness benefit are all the same. No extra payments payable with JB and IB as they are considered short term payments. Fuel allowance may be payable with OPFP, subject to other condition...It will not be payable with half rate JB/IB.
HTH, A