Why is the minipage indented on the left?

This avoids the problem, I think:

%-------------------------------------------------------------------
% PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%-------------------------------------------------------------------
\documentclass[10pt]{article}
\usepackage{array, xcolor, lipsum, bibentry}
\usepackage[
    paper=letterpaper,
    includefoot,            % Uncomment to put page number above margin
    marginparwidth=1in,     % Length of section titles
    marginparsep=.05in,     % Space between titles and text
    margin=1in,         % 1 inch margins
    ]{geometry}

\usepackage{showframe}

% Shrink spacing around section headings
\usepackage{titlesec}
\titleformat*{\section}{\large\bfseries}
\titlespacing\section{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt}

% Break tables across pages
\usepackage{supertabular}

% Set column sizes and color
\definecolor{lightgray}{gray}{0.8}
\newcolumntype{L}{>{\raggedleft}p{0.1225\textwidth}}
\newcolumntype{R}{p{0.81\textwidth}}
\newcolumntype{X}{>{\raggedleft}p{0.05\textwidth}}
\newcolumntype{Y}{p{0.9\textwidth}}

% Commands to simplify starting a table
\newcommand\VRule{\color{lightgray}\vrule width 0.5pt}
\newcommand\smalltab{\begin{supertabular}{L!{\VRule}R}}
\newcommand\medtab{\begin{supertabular}{X!{\VRule}Y}}
\newcommand\longtab{\begin{supertabular}{L!{\VRule}R}}

\begin{document}
%-------------------------------------------------------------------
% NAME AND CONTACT INFO
%-------------------------------------------------------------------

\hspace*{-\parindent}%
\begin{minipage}[b]{0.65\textwidth}
\begin{flushleft}
    \Huge{\textsc{Here is my name\\}}
    \Large{Curriculum Vitae}
\end{flushleft}
\end{minipage}%
\begin{minipage}[b]{0.35\textwidth}
    \small{Street Address\\
    City, State 12345\\
    [email protected]\\
    www.website.com\\
    1-555-555-5551}
\end{minipage}

%-------------------------------------------------------------------
% EDUCATION
%-------------------------------------------------------------------
\section*{Education}
\smalltab
    2011--present & PhD in Stuff, Very Awesome School.\\[2.5pt]
    2009--2011 & Masters in Stuff, Just OK School.\\[2.5pt]
    2004--2008 & Bachelors in Things, Relatively Awesome School.\\
\end{supertabular}

\end{document}

Basically, TeX is treating the first minipage as the beginning of a paragraph and indenting it accordingly. Putting in negative space equal to the paragraph indentation fixes the problem:

Use \noindent:

\noindent
\begin{minipage}[b]{0.65\textwidth}

Thanks Charles Staats