sectsty only updates only the sectional unit font-related setting but still use the traditional \@startsection construction:
\renewcommand\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
% {\normalfont\Large\bfseries}}
{\normalfont\Large\bfseries\SS@sectfont}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
% {\normalfont\large\bfseries}}
{\normalfont\large\bfseries\SS@subsectfont}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subsubsectfont}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
{3.25ex \@plus1ex \@minus.2ex}%
{-1em}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@parafont}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{3.25ex \@plus1ex \@minus .2ex}%
{-1em}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subparafont}}
The normal font-selections are commented out and substituted with the sectsty formatting \SS@.... Read up on the meaning of each component of \@startsection in Where can I find help files or documentation for commands like \@startsection for LaTeX? and focus on arguments 4 (before skip) & 5 (after skip).
Here's an example that adjust the space before and after the \subparagraph unit:

\documentclass{article}
\usepackage{sectsty}
\allsectionsfont{\centering\normalfont\normalsize}
\subsectionfont{\noindent\normalfont\normalsize\emph}
%\subsubsectionfont{\indent\normalfont\normalsize\itshape}
\subparagraphfont{\indent\normalfont\normalsize\itshape}
\begin{document}
\section{A section}
Some text
\subsection{A subsection}
Some text
\subsubsection{A subsubsection}
Some text
\paragraph{A paragraph}
Some text
\subparagraph{A subparagraph}
Some text
\makeatletter
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
{2\baselineskip \@plus1ex \@minus .2ex}%
{\baselineskip}%
% {\normalfont\normalsize\bfseries}}
{\normalfont\normalsize\bfseries\SS@subparafont}}
\makeatother
\subparagraph{A subparagraph}
Some text
\end{document}
Note that sectsty is assuming that your working with a document class for which the sectional units use \@startsection without modification from the default classes, with the exclusion of KOMA-Script (from the sectsty documentation; section 1 Introduction, p 2):
The secsty package provides a set of commands for changing the font
used for the various sectional headings in the standard LaTeX 2e document
classes: article, book, and report. This package also works with the
KOMA-Script classes scrartcl, scrbook, and scrreprt.
While it "will work" with other document classes, the effect is merely to overwrite the sectional units' definitions with that of the default document classes mentioned above. For example, apa6 defines
\renewcommand{\section}{\@startsection {section}{1}{\z@}%
{\b@level@one@skip}{\e@level@one@skip}%
{\centering\normalfont\normalsize\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{\z@}%
{\b@level@two@skip}{\e@level@two@skip}%
{\normalfont\normalsize\bfseries}}
\newcommand*{\addperi}[1]{#1.}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{\parindent}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries\addperi}}
\renewcommand{\paragraph}{\@startsection{paragraph}{4}{\parindent}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries\itshape\addperi}}
\renewcommand{\subparagraph}[1]{\@startsection{subparagraph}{5}{1em}%
{0\baselineskip \@plus 0.2ex \@minus 0.2ex}%
{-\z@\relax}%
{\normalfont\normalsize\itshape\hspace{\parindent}{#1}\textit{.}}{\relax}}
In order to maintain the sectional unit structure, you're better off redefining the above set manually* (separate from sectsty, as was done in the first example). The same approach would hold (including a redefinition to your liking within a \makeatletter...\makeatother pair) as apa6 also uses \@startsection to define the sectional units.
* Of course, other packages could be used as well, if they are compatible with apa6, but that seems to be out of the question here.
apa6, I'll post a complete example when I have access to the original document. – alfC Mar 29 '14 at 07:50