How can I align at relation symbols within a cases environment?

You can use the align environment, which is provided by the amsmath package. I see you load it anyway:

\documentclass{article}
\usepackage{amssymb,amsmath}
\begin{document}

\begin{table}[H]
\centering
  \begin{tabular}{c|c}
  \hline 
  { $X=y+z$}  & { 
   \parbox[c]{0.75\columnwidth}{
   \begin{align}
   C_L&= \begin{cases} 
        min\left[\left(Re\right),\: f(B)\right]\:;\: A<4\\   
        f(E)\:;\:4\le E<10\\
         -0.29\:;\: E\ge10
        \end{cases}\\ 
    f(E)&= E^{3}-0.0159E^{2}-0.0204E+0.474
   \end{align}
  }
}\tabularnewline \hline
\end{tabular} 
\end{table}

\end{document}

The ampersands before the equal signs set the characters that need to be aligned, meaning that if you used it in another place, the character that follows will be aligned.

And here's the output of the code:

You can push the cases content to the right the same amount as f(E) using

$\phantom{f(E)}\llap{$C_L$} = \begin{cases}
%...

The left overlap makes a right-aligned, zero-width box to not let C_L influence the spacing.

Ps. You may want to use \min instead of just typing min in math mode.

use of the aligned environment seems the simplest fix to me:

\documentclass{article}
\usepackage{amssymb,amsmath}
\begin{document}

\begin{table}[H]
\centering
  \begin{tabular}{c|c}
  \hline 
  { $X=y+z$}  & { 
   \parbox[c]{0.75\columnwidth}{
   \( \begin{aligned}
   C_L&= \begin{cases} 
        min\left[\left(Re\right),\: f(B)\right]\:;\: A<4\\   
        f(E)\:;\:4\le E<10\\
         -0.29\:;\: E\ge10
        \end{cases}\\ 
    f(E)&= E^{3}-0.0159E^{2}-0.0204E+0.474
   \end{aligned} \)
  }
}\tabularnewline \hline
\end{tabular} 
\end{table}

\end{document}

two things to notice:

• within a \parbox, the aligned environment needs to be put explicitly into math mode;
• the original example had more space at the bottom than at the top; this was caused by an extra \\ after the last line, which has been removed here.