How to delete the space before and after the 'itemize' or 'enumerate'?

As @Bernard has already suggested in a comment, you may want to load the enumitem package, switch from itemize to enumerate environments, and start up both enumerate environments with the option nosep.

The following screenshot shows the outputs of both your original code and of the code that implements the preceding suggestions.

documentclass{article}
usepackage{enumitem}
setlengthparindent{0pt}
begin{document}
% Before:
hrule
textbf{I. Introduction}

hspace*{1em}textbf {A. Background:}
begin{itemize}
setlength{itemsep}{0pt}
setlength{parsep}{0pt}
setlength{parskip}{0pt}
item[1.] The differences between clear vowels and conversational 
          vowels are widely studies.
begin{itemize}
item[a.]Vowel Duration: Clear vowels are generally longer than 
         conversational vowels
item[b.] Dynamic Formant Movement: Clear vowels exhibit greater 
         movement than naturally produced vowels.
end{itemize}
end{itemize}
hrule
% After:
textbf{I. Introduction}

hspace*{1em}textbf{A. Background:}
begin{enumerate}[nosep,label=arabic*.]
item The differences between clear vowels and conversational 
      vowels are widely studies.
   begin{enumerate}[nosep,label=alph*.]
   item Vowel Duration: Clear vowels are generally longer than 
         conversational vowels
   item Dynamic Formant Movement: Clear vowels exhibit greater 
         movement than naturally produced vowels.
   end{enumerate}
end{enumerate}
hrule
end{document}