In English it is most often better to be explicit than implicit. Saying "My father doesn't work today, because he is sick", somewhat implies that your father never works on this specific day. For a better example, If your father were to take the day off every Thursday, regardless of the circumstances, It would be appropriate to say, "My father doesn't work today, because he has the day off every Thursday". On the other hand, If he were to simply take the day off one Thursday because he couldn't make it in, it would be more appropriate to say, "My father isn't working today, because he is taking the day off". For one more example, if you meet someone who smokes cigarettes it would be appropriate to say that this person smokes. But if you meet someone who doesn't smoke cigarettes but is having just one cigarette at the moment, you would say this person is smoking.
Use doesn't when the situation is a recurring situation, and isn't when it is a one time situation that most likely will not repeat itself.