1. Try writing three different functions, one each for counting words, sentences, and paragraphs. You may have to duplicate some code, but by separating the the three tasks you will have an easier time to test for the correct conditions, and you will be able to solve one problem at a time.
2. It's a good idea to initialize variables when you define them, but you should use reasonable values. I understand your reasons to initialize some with 1 instead of 0, but it definitely looks odd, and makes it harder to understand your code and follow its logic.
3. Your test for blanks does not consider other cases of 'whitespace', such as tab characters, carriage return, form feed, or multiple whitespace characters. It may not apply to the case you are testing here, but if you base your code only on the specific test case you have, you may as well count the words by hand and return these numbers rather than write an entire algorithm around it...
4. '.' is not the only way to end a sentence. Also, depending on where you get your text from, you may be confronted with sequences of multiple punctuation marks!!! ;-)
5. The condition
if (userInput[i] == '\n' && userInput[i] == '\t')
is always false and always will be, no matter the text. Besides, why do you test for a tab character ( '\t' )? Reconsider the definition of paragraph that you use, or, rather, the definition of what separates paragraphs.
6. As a general rule, always consider corner cases: e. g. multiple separater characters where you only expect one, omitting or adding a separator at the end of the text, using variants on the commonly used separators, or interpreting characters that are not part of the readable text, but not one of the separators you catch either.