Choppy has listed a few of the usual ways to express the idea that your father is a businessman.
My father's occupation is a businessman....This is wrong. An occupation cannot be a person.
My father's occupation is businessman....correct though unusual.
If you consider businessman as a general term synonymous with the occupation it represents (business). then your sentence makes sense. Businessman here does not refer to an individual person; it refers to to all businessmen in general, who as a group represent the occupation. If used as a general term 'businessman' would NOT require an article.
Example
What a work of art is man. (Shakespeare)......"Man" is used as a general term for the entire human race
My father's occupation is "businessman".....If you put the word between quotation marks, to indicate that you are using the term in a special way, a grammar teacher would have no reason for objection. :)
More accurate would be...
My father's occupation is business.
My dentist's occupation is dentistry.