The following code snippet will guide you through the issue chronologically.
\documentclass[preview,border=12pt,12pt]{standalone}
\usepackage{mathtools}
\begin{document}
\section*{gather+aligned}
\begin{gather}
\left\{\!
\begin{aligned}
x &= \cos t\\
y &= \sin t
\end{aligned}\right.\\
x^2 +y^2 =1
\end{gather}
\section*{gather+split}
\begin{gather}
\left\{
\begin{split}
x &= \cos t\\
y &= \sin t
\end{split}\right.\\
x^2 +y^2 =1
\end{gather}
\section*{align+aligned}
\begin{align}
&\left\{\!
\begin{aligned}
x &= \cos t\\
y &= \sin t
\end{aligned}\right.\\
&x^2 +y^2 =1
\end{align}
\section*{align+split}
It cannot be compiled.
%\begin{align}
%&\left\{
%\begin{split}
%x &= \cos t\\
%y &= \sin t
%\end{split}\right.\\
%&x^2 +y^2 =1
%\end{align}
\section*{align+split+nobrace}
It can be compiled.
\begin{align}
&
\begin{split}
x &= \cos t\\
y &= \sin t
\end{split}\\
&x^2 +y^2 =1
\end{align}
\end{document}
Why can't split
with curly-brace \left\{
and \right.
be inside align
but it can be inside gather
?