memoir
defines a "parent" macro for each of the subfloat counters. More specifically, when you issue the command
\newsubfloat{figure}% Allow for subfigures
it creates the subfigure
counter as well as a p@subfigure
macro to extract/prints the parent figure number. Modifying this macro to be empty removes this functionality. If you only want to disable this "parent" counter removal temporarily, you should use grouping to limit the scope of the macro redefinition:

\documentclass{memoir}
\usepackage[margin=10mm]{geometry}% http://ctan.org/pkg/geometry
\usepackage[demo]{graphicx}% http://ctan.org/pkg/graphicx
% Allow subfloats for the following floats via \subbottom and/or \subtop
\newsubfloat{figure} % Subfigures
\newsubfloat{table} % Subtables
\begin{document}
\begingroup
\makeatletter
\renewcommand{\p@subfigure}{}% Void parent macro for figures
\makeatother
\begin{figure}[ht]
\centering
\subbottom[This is a subfigure\label{fig:label:a}]{\includegraphics{figure1}} \qquad
\subbottom[This is a subfigure\label{fig:label:b}]{\includegraphics{figure2}}
\caption{These are some awesome figures}
\label{fig:label1}
\end{figure}
See Figure~\ref{fig:label1}. It has subfigures~\ref{fig:label:a} and~\ref{fig:label:b}.
\endgroup
\begin{figure}[ht]
\centering
\subbottom[This is a subfigure\label{fig:label:c}]{\includegraphics{figure3}} \qquad
\subbottom[This is a subfigure\label{fig:label:d}]{\includegraphics{figure4}}
\caption{These are some awesome figures}
\label{fig:label2}
\end{figure}
See Figure~\ref{fig:label2}. It has subfigures~\ref{fig:label:c} and~\ref{fig:label:d}.
\begin{table}[ht]
\centering
\subbottom[This is a subtable\label{tbl:label:a}]{\includegraphics{table1}} \qquad
\subbottom[This is a subtable\label{tbl:label:b}]{\includegraphics{table2}}
\caption{These are some awesome tables}
\label{tbl:label}
\end{table}
See Table~\ref{tbl:label}. It has subtables~\ref{tbl:label:a} and~\ref{tbl:label:b}.
\end{document}
geometry
was loaded just to make the minimal example fit on a single page.
\documentclass
so that those trying to help don't have to recreate it.