Add variable=\y. The result is obviously the same.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[domain=0:2, scale = 0.75]
\draw(-4,0)--(4,0);
\draw (0,-4)--(0,4);
%\draw[black, line width = 0.50mm] plot[smooth,domain=-2:2] (\x, {(\x)^2});
\draw[black, line width = 0.50mm] plot[smooth,domain= -2:2,variable=\y] (\y, {(\y)^2});
\end{tikzpicture}
\end{document}
You could also call the plot variable \CuteFurryRodent, and you will still get
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[domain=0:2, scale = 0.75]
\draw(-4,0)--(4,0);
\draw (0,-4)--(0,4);
\draw[black, line width = 0.50mm]
plot[smooth,domain= -2:2,variable=\CuteFurryRodent] (\CuteFurryRodent, {(\CuteFurryRodent)^2});
\end{tikzpicture}
\end{document}

You may want to swap the roles of x and y,
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[domain=0:2, scale = 0.75]
\draw(-4,0)--(4,0);
\draw (0,-4)--(0,4);
\draw[black, line width = 0.50mm] plot[smooth,domain=-2:2] (\x, {(\x)^2});
\draw[blue,dashed, line width = 0.50mm] plot[smooth,domain= -2:2,variable=\y]
( {(\y)^2},\y);
\end{tikzpicture}
\end{document}

One could also use parabola bend here.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[domain=0:2, scale = 0.75]
\draw(-4,0)--(4,0);
\draw (0,-4)--(0,4);
\draw[black, line width = 0.50mm] (-2,4) parabola bend (0,0) (2,4);
\draw[blue,dashed, line width = 0.50mm,rotate=-90] (-2,4) parabola bend (0,0) (2,4);
\end{tikzpicture}
\end{document}