How do I position pairs of arrows in TikZ to make 'harpoons'?

I would suggest using the chemfig package for such purposes. It has built-in commands for schemes like these which are highly customizable. What's more it provides several arrow types, one of them being harpoons for an equilibrium reaction:

\documentclass{article}
\usepackage{chemfig,siunitx}
\setchemfig{compound sep=7em} % (not exactly) the length of the arrows
\begin{document}

\schemestart
  A \+ R
  \arrow{<=>[$k_{+1}$][$k_{-1}$]}
  ARb
  \arrow{<=>[$\alpha$][$\beta$]}
  ARc
  \arrow(ARc--){<=>[$\delta$][$\gamma$]}[35] AR$^*$
  \arrow(@ARc--){<=>[\SI{1000}{\per\second}][\SI{10}{\per\second}]}[-35] AD
\schemestop

\end{document}

In answer to my original question (above), I found that using the transform canvas command provided the easiest way to generate rightleftharpoon-like arrows using TikZ. Thanks to Caramdir for pointing out this previous answer. Below is a simple (somewhat meaningless example) showing both horizontal and angled harpoons.

\resizebox{!}{!}{  % picture placed inside a resize box (change to values etc as per desired sizing)

\begin{tikzpicture}

    \node (A) at (0,0) {A + R};
    \node (B) at (1.7,0) {AR};
    \node (C) at (3,1) {AR*};
    \node (D) at (3,-1) {AD};

\begin{scope}[every node/.style={font= \scriptsize}]

\draw[transform canvas={yshift=0.3ex},-left to] (A) -- node[above]{$k_{+1}$}  (B);
\draw[transform canvas={yshift=-0.3ex},left to-] (A) -- node[below] {$k_{-1}$} (B); 

\draw[transform canvas={yshift=0.2ex, xshift=-0.05ex},-left to] (B) --  node[above, xshift=-5]{$\beta$} (C);
\draw[transform canvas={yshift=-0.32ex, xshift=0.2ex},left to-] (B) --  node[below, xshift=5]{$\alpha$} (C);

\draw[transform canvas={yshift=0.32ex, xshift=0.2ex},-left to] (B) --  node[above, xshift=15]{1000 s$^{-1}$}  (D);
\draw[transform canvas={yshift=-0.2ex, xshift=-0.05ex},left to-] (B) --  node[below, xshift=-10]{10 s$^{-1}$}  (D);

\end{scope}

\end{tikzpicture} 

}

This produces:

\documentclass{minimal}
\usepackage{mathtools}

\begin{document}    
\[
 A + R \xrightleftharpoons[k_{-1}]{k_{+1}} AR \xrightleftharpoons[\alpha]{\beta} AR*
\]
\end{document}