To get Disabilty Allowance, a person has to have a medical condition that is expected to last at least a year. This is very vague. In my experience, a person can have an illness and the effects of that illness (not necessarily the illness itself) can last for more than a year. I'm sorry if this sounds convuluted.
Have a look at this link and search for Disability Allowance -
http://www.welfare.ie/EN/OperationalGuidelines/Pages/medreview.aspx
The relevant text is:
'Disability Allowance:
The Medical Assessor will give an opinion on whether the person, by reason of a specified disability, may or may not be considered substantially handicapped in undertaking work of a kind which, if they were not suffering from that disability would be suited to their age, experience and qualifications.
This opinion is referred to the Deciding Officer for a decision on the claim (means criteria must also be satisfied for Disability Allowance).'
There's a huge waiting list for Disabilty Allowance (DA) and Carer's Allowance (CA). There is nothing preventing your dad from applying for Jobseeker's Allowance (which if granted, would include a payment for your mum') and your mum applying for DA.
If your dad applied for Carer's Allowance, a Deciding Officer may consider that your dad is not available for full-time work as he has submitted a CA claim (it would be seen on the computer system).
In my view there are two options open to your parents.
1. Your dad applies for Jobseeker's Allowance for both of them. As this is a lengthy process he should apply for Supplementary Welfare Allowance while the JA claim is decided.
2. Your dad applies for CA and your mum applies for DA. Again he should apply for Supplementary Welfare Allowance. This is paid while claims are pending.
The only reason I suggested the CA and DA option is because of the extra benefits associated with CA, such as, Free Travel, Household Benefits Package and the Respite care Grant.
No matter what option they choose, they should apply for Supplementary Welfare Allowance. This is a weekly payment from the Community Welfare Officer, usually at their local health centre.
http://www.welfare.ie/EN/OperationalGuidelines/Pages/swa.aspx
Best wishes to your parents (especially your mum with her future treatment).