UPDATE (08/03/18): This issue has been fixed in the latest version of the tikzmark package. The workaround proposed by cfr, using \beameroriginal to redefine the \tikzmark inside beamer environments, has now been incorporated into the tikzmark package. So this question now serves only a historical purpose.
Formerly, this worked, including with pgf version 3.0.1a:
\documentclass{beamer}
%\documentclass{article}
\usepackage{tikz,pgfplots}
\usetikzlibrary{positioning,calc,tikzmark}
\begin{document}
\begin{frame}\frametitle{A SLIDE}
meaningless filler\tikzmark{a}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
% old preferred syntax for using tikz marks inside tikz pictures
\node (foo) at (1,-3) {\tikzmark{b}};
% new preferred syntax for using tikz marks inside tikz pictures
% \tikzmark{b}{(1,-3)}
\end{tikzpicture}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
\draw[black] (pic cs:a) -- (4,-3);
\draw[black] (pic cs:b) -- (4,-3);
\end{tikzpicture}
\end{frame}
\end{document}
Now I've updated some packages--presumably including beamer--and it fails with the error "Cannot parse this coordinate." A little searching the interweb shows that the preferred way to use \tikzmark inside a tikzpicture is now \tikzmark{b}{(1,-3)}. Yet this fails also with the same error:
\documentclass{beamer}
%\documentclass{article}
\usepackage{tikz,pgfplots}
\usetikzlibrary{positioning,calc,tikzmark}
\begin{document}
\begin{frame}\frametitle{A SLIDE}
meaningless filler\tikzmark{a}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
% old preferred syntax for using tikz marks inside tikz pictures
% \node (foo) at (1,-3) {\tikzmark{b}};
% new preferred syntax for using tikz marks inside tikz pictures
\tikzmark{b}{(1,-3)}
\end{tikzpicture}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
\draw[black] (pic cs:a) -- (4,-3);
\draw[black] (pic cs:b) -- (4,-3);
\end{tikzpicture}
\end{frame}
\end{document}
On the other hand, the problem seems to be local to beamer, because this works:
%\documentclass{beamer}
\documentclass{article}
\usepackage{tikz,pgfplots}
\usetikzlibrary{positioning,calc,tikzmark}
\begin{document}
%\begin{frame}\frametitle{A SLIDE}
meaningless filler\tikzmark{a}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
% old preferred syntax for using tikz marks inside tikz pictures
% \node (foo) at (1,-3) {\tikzmark{b}};
% new preferred syntax for using tikz marks inside tikz pictures
\tikzmark{b}{(1,-3)}
\end{tikzpicture}
\begin{tikzpicture}[overlay,>=latex,shorten >=1pt,->,remember picture]
\draw[black] (pic cs:a) -- (4,-3);
\draw[black] (pic cs:b) -- (4,-3);
\end{tikzpicture}
%\end{frame}
\end{document}
(And finally, for completeness, using \node (foo) at (1,-3) {\tikzmark{b}}; in the article class, but with the new packages, also fails, perhaps as expected.)
I dread trawling through the beamer class files. Is there something "obvious" I'm missing first?

rememberand/oroverlayseems deeply problematic. – cfr Feb 21 '17 at 23:39overlayorremember picture: the error still occurs without them, and does not occur under thearticledocumentclass. – Quiggin Feb 21 '17 at 23:46\tikzmarkis not the same as\tikzmarkusually is. Otherwise, it would not be overlay aware. Theremember picturejust seems pointless. But theoverlaydoesn't seem coherent. I don't care if it works - it is completely illogical! – cfr Feb 22 '17 at 00:42