If you know which newspaper he is reading (it is the same newspaper every day) you can use the definite article. The newspaper is defined:
My father is reading the newspaper.
If you do not have a particular newspaper in mind, you are just announcing that he is in the process of reading and it is a newspaper (and not a book for example), then you can use the indefinite article:
My father is reading a newspaper.
The last sentence is not correct because "newspaper" is a singular countable noun and it needs to be preceded by an article (or a possessive adjective - my, your, etc.)
A singular noun on its own is used without an article for example when it is uncountable: My father is drinking milk.