Why are you using
Applet? Applet has been deprecated since version 9.
You only need one instance of Random. You could make it a private static final Random just inside the class declaration.
Your various arrays for parts of speech should also be make private static final, i.e. as constants.
Your indentation needs a lot of clean up.
You are always choosing the range of the random number based on the SubjectForSentence.length. Where you are using the array applies to arrays of differing lengths.
You are creating a
String array for StoreSentences but initializing it to a length of zero.
You need to create some more methods because the body of Paragraph() is way too big. And method names should begin with a lower case character, so, paragraph().
How about
And,
Where you build the sentence string you add in a bunch of "". That's an empty string and serves no purpose they way you've used it.