Can your dad able to move or no movement at all ? On permanent or temporary tube feeding or urinal tube (these will need professional nursing care if permanently) ?
Staying in nursing home he will likely deteriorated and lost all function or 6 ADL very fast as exercise and movement is miniment. Always on bed due to fall risk like in hospital (Non compliance will tie to bed, very sad). Nurse or nursing aide are limited and the ratio to nursing home resident very low or tight. It's not 1-1 or 1-2 ratio type, so cannot expect constant assistance or support like maid or helper. Nursing home likely only change diapers on fixed timings, food feeding, bed bathing or wheelchair shower, and don't expect to help dad assisted walking as when he like or push him wheelchair see scenary etc. Nursing home physiotherapy is also group. It can be as short as 10-30mins and back to bed. Nothing much and useless likely. If bed bound, turning body also not frequent or not at all.
It's like waiting to rot and die if stay into nursing home.
Getting a helper will be financially cheaper with closer care (your mum can see your dad daily). Get one who willing to take care your dad. The rest depends on luck and no point thinking too much. Buy portable CCTV and place them to view house and your parents conditions. Explore with hospital social worker home care help and financial $$ support she can advise of provide like daycare centre for exercise for stroke recovery, depending on the stroke conditions.
Anything emergency, just direct call ambulance send hospital.
6 ADLs:
They include bathing or showering, dressing, getting in and out of bed or a chair, walking, using the toilet, and eating