Marking a point on parabola (like ellipse)

Question

We can mark a point on ellipse using

\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-120:2cm and 1cm) {};

Is there any way to define a point on parabola?

\documentclass[border=2pt]{standalone}
\usepackage[usenames,dvipsnames,svgnames]{xcolor}
\usepackage{tikz}

\color{Maroon} 

\begin{document}
\begin{tikzpicture}
\draw[thin,gray!50](-3,-2) grid (3,2);
\draw[Maroon,thick] (0,0) ellipse (2cm and 1cm);
\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-120:2cm and 1cm) {};

\draw[red] (-1,0) parabola[parabola height=1.5cm] (2,0);
\end{tikzpicture}

\end{document}

Answer 1

Based on marmot comment...

\documentclass[border=2pt]{standalone}
\usepackage[usenames,dvipsnames,svgnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\color{Maroon} 

\begin{document}
\begin{tikzpicture}
\draw[thin,gray!50](-3,-2) grid (3,2);
\draw[Maroon,thick] (0,0) ellipse (2cm and 1cm);
\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-150:2cm and 1cm) {};

%\draw[red] (-1,0) parabola[parabola height=1.5cm] (2,0);
\draw[red,postaction={decorate,decoration={markings, mark=at position 0.5 with { \node[red,draw,fill,circle,inner sep=1pt,label={-90:$Y$}](Y){};}}}] (-1,0) parabola[parabola height=1.5cm] (2,0);
\end{tikzpicture}

\end{document}