No, unfortunately, you may not. The requirements for continued eligibility for unemployment benefits are clear: you must continue looking for work (and, if you receive a job opportunity at all comparable in level or salary to your prior work, take it). There is no exception allowing you to not look for work to care for a family member. Just as, if you were still employed, your employer would not have to pay you if you took time off from work  to care for your father (obviously, unless you had and used PTO that you'd earned), so, too you are not entitled to unemployment benefits unless you continue to at least make the effort to work.