There's probably a package to do this ... but in the meantime, here's a way to do it that keeps the baseline correct. It lowers the line by a specified amount just before the opening parenthesis and then raises it back to where it should be just after. Then it does the same again the other end. One could add additional options to have one command deal with all styles of parenthesis and to deal with inline versus displaystyle (this is for displaystyle) - I'd wait to see if there's a package before tackling that! The drawback of this is that you have to decide what the "drop" should be for yourself. A more sophisticated approach would compute that for you.
\documentclass{article}
\usepackage{amsmath}
\newcommand{\lowerparen}[2]{%
\raisebox{-#1}{\(\displaystyle\left(\raisebox{#1}{\(\displaystyle #2\)}\right)\)}}
\begin{document}
\[
Y = \lowerparen{6pt}{\sum_{\substack{0 \le i \le m \\ 0 < j < m}}X}
\]
\end{document}
Here's a slightly fuller solution, based on the above but with a little more flexibility with regard to mathmodes (thanks to this question on mathmodes).
\documentclass{article}
\newlength{\parenheight}
\newlength{\parendepth}
\newlength{\parendrop}
\newcommand{\paren}[4]{%
\settoheight{\parenheight}{\(#4 #2\)}%
\settodepth{\parendepth}{\(#4 #2\)}
\addtolength{\parendepth}{.5ex}
\addtolength{\parenheight}{-.5ex}
\addtolength{\parenheight}{\parendepth}
\addtolength{\parendepth}{-.5\parenheight}
\setlength{\parendrop}{-.5\parenheight}
\addtolength{\parendrop}{.5ex}
\raisebox{-\parendepth}{\(#4
\left#1%
\rule[\parendrop]{0pt}{\parenheight}%
\right.\)}
#2
\raisebox{-\parendepth}{\(#4
\left.%
\rule[\parendrop]{0pt}{\parenheight}%
\right#3\)}
}
\def\myleft#1#2\myright#3{%
\mathchoice{%
\paren{#1}{#2}{#3}{\displaystyle}%
}{%
\paren{#1}{#2}{#3}{\textstyle}%
}{%
\paren{#1}{#2}{#3}{\scriptstyle}%
}{%
\paren{#1}{#2}{#3}{\scriptscriptstyle}%
}%
}
\begin{document}
\(
\myleft(\prod_{{s = 0 \atop s \ne 3}} X_s\myright)^2 \left(A\right) \left(x\right)
\)
\[
\myleft(\prod_{{s = 0 \atop s \ne 3}} X_s\myright)^2 \left(A\right) \left(x\right)
\]
\end{document}
I haven't tested it very much. I suspect that something will break with the way I've coded the 'left-right' matching. The main point is to get the heights lined up.
(As this answer has been accepted, I don't feel I should simply replace the original answer with this new one but it's only a little development of the original so doesn't really deserve a new answer. So I'm posting it as an addendum.)
\Big(
. – Watson Jan 12 at 15:20