Vertical center alignment of text in cell of table

With use of the nccmath, cellspace, and hhline packages and array for table:

\documentclass{article}
\usepackage{nccmath}
\usepackage[table]{xcolor}
\usepackage{cellspace, hhline}
    \setlength\cellspacetoplimit{5pt}
    \setlength\cellspacebottomlimit{5pt}

\begin{document}
\[
    \begin{array}{ |c | c | >{$}Sc<{$}|}
    \hhline{---}
\rowcolor{blue!50} 
\text{\textcolor{white}{Parabola}} 
        & \text{\textcolor{white}{Equation of Tangent}} 
                & \text{\textcolor{white}{Point of Contact}}   \\
    \hhline{---}
y^2=4ax
    & y=mx+\frac{a}{m}
        & \Bigl(\mfrac{a}{m^2},\mfrac{2a}{m}\Bigr)      \\  \hhline{---}
y^2=-4ax
    & y=mx-\dfrac{a}{m}
        & \Bigl(-\mfrac{a}{m^2},\mfrac{2a}{m}\Bigr)     \\  \hhline{---}
x^2=4ay
    & y=mx-am^2
        & \bigl(2am,am^2\bigr)                          \\ \hhline{---}
{x^2=-4ay}
    & y=mx+am^2
        & \bigl(-2am,-am^2\bigr)                        \\ \hhline{---}
    \end{array}
\]
\end{document}

Addendum: Version where equation in cells are aligned at equal sign and points at commas:

\documentclass{article}
\usepackage{nccmath}
\usepackage[table]{xcolor}
\usepackage{cellspace, hhline}
    \setlength\cellspacetoplimit{5pt}
    \setlength\cellspacebottomlimit{5pt}
\usepackage{xparse}
\NewExpandableDocumentCommand\mcw{O{>{\color{white}}Sc|}m}
    {\multicolumn{2}{#1}{#2}}
\usepackage[skip=1ex]{caption}

\begin{document}
\[
\setlength\arrayrulewidth{0.5pt}
\setlength\arraycolsep{12pt}
    \begin{array}{ |r@{\,}l | >{\qquad}r@{\,}l | r@{,\,}>{$}Sl<{$}|}
    \hhline{*{6}{-}}
\rowcolor{blue!50}
\mcw[|>{\color{white}}Sc|]{Parabola}
        & \mcw{Equation of Tangent}
                & \mcw{Point of Contact}             \\
    \hhline{*{6}{-}}
y^2 & = 4ax
    & y & =mx+\frac{a}{m}
        & \Bigl(\mfrac{a}{m^2} & \mfrac{2a}{m}\Bigr)    \\  
    \hhline{*{6}{-}}
y^2 & = -4ax
    & y & = mx-\dfrac{a}{m}
        & \Bigl(-\mfrac{a}{m^2} & \mfrac{2a}{m}\Bigr)   \\  
    \hhline{*{6}{-}}
x^2 & = 4ay
    & y & = mx-am^2
        & \bigl(2am & am^2\bigr)                        \\ 
    \hhline{*{6}{-}}
x^2 & = -4ay
    & y & = mx+am^2
        & \bigl(-2am & -am^2\bigr)                      \\ 
    \hhline{*{6}{-}}
    \end{array}
\]
\end{document}

I suggest to use {NiceTabular} of nicematrix which provides parameters cell-space-bottom-limit and cell-space-top-limit inspired by the lengths \cellspacebottomlimit and \cellspacetoplimit of the package cellspace.

{NiceTabular} is designed to give better results than standard colortbl in the PDF viewers when rules are used in conjonction with colored panels. However, you need several compilations.

\documentclass{article}

\usepackage{nicematrix}

\begin{document}
\begin{center}
\everymath{\displaystyle}
\NiceMatrixOptions{cell-space-top-limit=2pt,cell-space-bottom-limit=2pt}
\begin{NiceTabular}{ccc}[hvlines,colortbl-like]
\rowcolor{blue!50} \textcolor{white}{Parabola} & \textcolor{white}{Equation of Tangent} & \textcolor{white}{Point of Contact} \\ 
$y^2=4ax$& $y=mx+\frac{a}{m}$& $\biggl(\frac{a}{m^2},\frac{2a}{m}\biggr)$  \\ 
$y^2=-4ax$ & $y=mx-\frac{a}{m}$ & $ \biggl(-\frac{a}{m^2},\frac{2a}{m}\biggr)$  \\  
$x^2=4ay$ & $y=mx-am^2$& $\bigl(2am,am^2\bigr)$  \\ 
$x^2=-4ay$ & $y=mx+am^2$& $\bigl(-2am,-am^2\bigr)$  \\   
\end{NiceTabular}
\end{center}
\end{document}