How do you make parentheses match height when they're split between lines in LaTeX math?

You should use \vphantom, it makes a vertical space equal to its argument and no horizontal space:

\begin{equation}
    \begin{split}
        f = & \left( \frac{a}{b} + \right. \\
                   & \left. \vphantom{\frac{a}{b}} c \right) + d 
    \end{split}
\end{equation}

(I recommend \vphantom over \phantom in this case because \phantom adds horizontal space that you don't need.)

For a lot of great advice on typesetting mathematics, have a look at Math mode by Herbert Voß.

Oh. It's the \phantom{} command

\begin{equation}
    \begin{split}
        f = & \left( \frac{a}{b} + \right. \\
                   & \left. \phantom{\frac{a}{b}} c \right) + d 
    \end{split}
\end{equation}

You can also use \big( \big) to set the size specifically.

The order goes (, \big(, \Big(, \bigg(, \Bigg( for increasing bigness.