Is it permissible to take a portion of the wealthy, sick father's money—whose care requires full-time dedication—to hire a worker for the commercial shop, which is the sole source of livelihood, with the intention of returning the money later?

It is not permissible to take anything from the father's money except with his permission, whether it be by way of borrowing or otherwise. Therefore, one must honor and be kind to the father as much as possible, in a way that does not harm the son or those he supports.