How to reduce space before minipage

In your set-up there is a \parskip before the minipage and once the minipage starts there is an \abovedisplayskip at the top each set of equations. You can undo either of these vertical spaces by issuing \vspace{-\parskip} or \vspace{-\abovedisplayskip} just before the minipage. Which one you choose is up to what spacing you wish to have:

\documentclass[12pt, letterpaper]{article}

\usepackage[a4paper,top=1 in,bottom=1 in,left=0.7 in,right=0.7 in]{geometry}

\usepackage[utf8]{inputenc}
\usepackage[misc]{ifsym}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{forest}
\usepackage{tikz}
\usetikzlibrary{intersections}

\usepackage[parfill]{parskip}
\setlength{\parskip}{1ex}

\begin{document}
\textbf{Answer:} Let the points be $A = (x_1 , x_2)$, $B = (y_1 ,
y_2)$ and $C = (z_1 , z_2)$.  Remove \verb+\parskip+.

\vspace{-\parskip}
\begin{minipage}[b]{.5\textwidth}
    \begin{flalign*}
        \rightarrow AB^2&=&\\
        &=&\\
        \therefore\;AB&=&
    \end{flalign*}
\end{minipage}%
\begin{minipage}[b]{0.5\textwidth}
    \begin{flalign*}
        \rightarrow BC^2&=&\\
        &=&\\
        \therefore\;BC&=&
    \end{flalign*}
\end{minipage}

\textbf{Answer:} Let the points be $A = (x_1 , x_2)$, $B = (y_1 ,
y_2)$ and $C = (z_1 , z_2)$.  Remove \verb+\abovedisplayskip+.

\vspace{-\abovedisplayskip}
\begin{minipage}[b]{.5\textwidth}
    \begin{flalign*}
        \rightarrow AB^2&=&\\
        &=&\\
        \therefore\;AB&=&
    \end{flalign*}
\end{minipage}%
\begin{minipage}[b]{0.5\textwidth}
    \begin{flalign*}
        \rightarrow BC^2&=&\\
        &=&\\
        \therefore\;BC&=&
    \end{flalign*}
\end{minipage}

\textbf{Answer:} Let the points be $A = (x_1 , x_2)$, $B = (y_1 ,
y_2)$ and $C = (z_1 , z_2)$.  Remove nothing.

\begin{minipage}[b]{.5\textwidth}
    \begin{flalign*}
        \rightarrow AB^2&=&\\
        &=&\\
        \therefore\;AB&=&
    \end{flalign*}
\end{minipage}%
\begin{minipage}[b]{0.5\textwidth}
    \begin{flalign*}
        \rightarrow BC^2&=&\\
        &=&\\
        \therefore\;BC&=&
    \end{flalign*}
\end{minipage}

\end{document}