I am not an expert on this, but to get the thread going.
It appears that there are three groups for Universal Credit.
People who are deemed to be fit for work, they need to either be in work, or carrying out work-related activity, which, I think but I am not certain, includes applying for work, carrying out training.
People who are not able to work at present but must be carrying out work-related activity.
People who are long-term sick or disabled, who are not required to do work or work-related activity but are required to "sign on" or whatever they call it now.
There has been some discussion on one of the threads to which i linked as to the jobcentre expecting someone who can hardly walk to "pop in" to the jobcentre to sign, but in that particular case a member of jobcentre staff did it over the telephone for her.
I am no expert on this, but in the hope that others might join in the discussion, it may be that the best approach to the situation is that your mother is a full-time carer for your father.
The system may have changed, but it used to be that depending upon his situation your father woud get Attendance Allowance and/or your mother would get Carer's Allowance.
LINK > www.gov.uk/attendance-allowance
LINK > www.gov.uk/carers-allowance
Some of this will depend if your father is over State Pension Age, and upon what is his medical situation.
I do not know if this links in to Universal Credit in any way, or if it is entirely separate from Universal Credit.
Hopefully others will join in this discussion.
I hope this helps.